by Yuguang (Roger) Bai

Cluster Structure for Mirkovic-Vilonen Cycles and Polytopes

by

Yuguang (Roger) Bai

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

c© Copyright 2021 by Yuguang (Roger) Bai

Abstract

Cluster Structure for Mirkovic-Vilonen Cycles and Polytopes

Yuguang (Roger) Bai

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2021

ii

Acknowledgements

iii

Contents

1 MV Basis 6

1.1 MV Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 MV Basis of C[N ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Product of MV Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Semicanonical Basis 10

2.1 Preprojective Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Semicanonical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Polytopes 14

3.1 Polytopes from MV Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.1 BZ Datum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.2 Lusztig Datum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Polytopes from Λ-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Cluster Algebras 19

4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Cluster Structure for C[N ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Cluster Structure for Λ-mod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.4 Lusztig Datum and g-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Cluster Structure for MV Polytopes 30

5.1 Exchange Relation coming from Λ-mod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 Exchange Relation Coming from MV Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Investigating the Cluster Structure for MV Cycles 35

6.1 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.2 Terms in the Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7 Fusion Product in Type A 47

7.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.1.1 Rings and discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.1.2 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.1.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

iv

7.2 Affine Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7.2.1 Relations between different affine Grassmannians . . . . . . . . . . . . . . . . . . . 49

7.2.2 The fusion construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7.2.3 Some subschemes of affine Grassmannians . . . . . . . . . . . . . . . . . . . . . . . 50

7.3 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.3.1 Adjoint orbits and their deformations . . . . . . . . . . . . . . . . . . . . . . . . . 52

7.3.2 The Mirkovic–Vybornov slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7.4 Mirkovic–Vybornov Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7.4.1 Linear Operators from lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7.4.2 Matrices from lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.4.3 Upper-triangularity and the Mirkovic-Vybornov isomorphism . . . . . . . . . . . . 56

7.5 Application to the Fusion Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.5.1 MV cycles and tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.5.2 Fusion of MV cycles via fusion of generalized orbital varieties . . . . . . . . . . . . 60

7.5.3 Multiplying MV basis elements in C[N ] . . . . . . . . . . . . . . . . . . . . . . . . 62

7.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

8 A3 Case Study 66

8.1 Clusters in C[N ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

8.2 Clusters in Λ-mod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

8.3 The MV Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

8.4 The MV Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Bibliography 85

v

Introduction

Canonical Bases

For a simple simply-connected complex algebraic group G with unipotent N ⊂ G, there have been a

number of bases introduced for C[N ] that have several nice properties. For instance, as it is possible to

view every irreducible representation L(λ) of G as a subspace of C[N ], we would like a basis β ⊂ C[N ]

such that β ∩ L(λ) is a basis for each L(λ).

Such bases include:

1. Lusztig’s dual canonical basis [Lus90]

2. Lusztig’s dual semicanonical basis [Lus00]

3. The MV basis [BKK19]

These bases are in general different from each other, and there is a well-known type A5 example showcas-

ing this [GLS05][BKK19]. As these bases are very useful, one would like to compute their basis elements.

However, this computation is in general geometric and difficult. For example, the dual canonical basis

comes from perverse sheaves, the dual semicanonical basis is derived from constructible functions on

irreducible components of varieties of modules, and the MV basis originates from cohomology classes.

As such, we would like a combinatorial way to write down all basis elements. A partial answer to this

problem can be achieved through cluster algebras.

Cluster Algebras

Cluster algebras were introduced by Fomin and Zelevinsky in [FZ02] and it was shown in [GLS07a], using

the construction of [BFZ05], that C[N ] has a natural cluster algebra structure. This structure is described

axiomatically from an initial set of generators, called an initial seed, which is a pair Σ = ((x1, . . . , xr), B)

with B being a matrix. We then obtain new generators through a process called mutation, where for

each mutable xi, there is a unique element x∗i satisfying an equation of the form

xix∗i = x+ + x−.

This equation is called an exchange relation and the terms x+ and x− are certain products of the xj ,

j 6= i, determined by B. We then acquire a new seed

µi(Σ) = ((x1, . . . , xi−1, x∗i , xi+1, . . . , xr), µi(B))

where µi(B) is another matrix derived from B.

1

2

We call each of the variables x1, . . . , xr and all variables that are produced through successive muta-

tions cluster variables. As computation of these cluster variables is combinatorial in nature, a natural

question is whether or not the cluster variables are elements of the aforementioned bases. If so, then a

large number of basis elements can be efficiently computed.

For type ADE, it has been shown in [KKKO18] that the cluster monomials, products of cluster

variables that all belong to a single seed, belong to the dual canonical basis and in [GLS06] for the dual

semicanonical basis. The question is still open for the MV basis, but in [BGL20], the authors showed

that for certain seeds satisfying a special condition, the cluster monomials in those seeds belong to the

MV basis. Furthermore, in type An, n ≤ 3, the MV basis agrees with the dual canonical and dual

semicanonical basis [BKK19] so the cluster monomials are in the MV basis in those cases. This leads to

the main conjecture this thesis attempts to investigate:

Conjecture. The cluster monomials in C[N ] belong to the MV basis.

Cycles, Modules, and Polytopes

Our approach to investigating the above conjecture will be relating the MV basis to the dual semicanon-

ical basis. The MV basis is created from and indexed by stable MV cycles, certain subvarieties of the

affine Grassmannian Gr. More specifically, there are certain orbits in Gr, denoted Grλ, Sµ−, and Sµ+, where

λ is a dominant weight and µ is a weight, and MV cycles are the irreducible components of Grλ ∩ Sµ−.

The stable MV cycles are the irreducible components of S0+ ∩ S

µ−.

The dual semicanonical basis is related to irreducible components of varieties of modules over the

preprojective algebra Λ and there is a bijection between MV cycles and certain Λ-modules called generic,

by first going through MV polytopes [BK10]. For an MV cycle Z, its MV polytope Pol(Z) can be thought

of as a moment polytope for Z [And03], and for a Λ-module M , its polytope Pol(M) is equal to the

convex hull of dimension vectors of submodules of M [BKT14]. Hence we have the following diagram:

C[N ] Generic Λ-modules

Stable MV Cycles MV Polytopes

(?)

For a stable MV cycle Z, let bZ ∈ C[N ] be the associated MV basis element. Each reduced expression

i of the longest word in the Weyl group of G gives us an initial seed Σ = ((x1, . . . , xr), B) of C[N ]

[BFZ05][GLS07a]. In this case, the cluster variables xi are in the MV basis [BKK19], say xi = bZi . We

are interested in the exchange relation xix∗i = x+ + x− where x∗i is the mutation of xi.

Fusion Product

There is a way to multiply MV cycles, called the fusion product and denoted by ∗, that is compatible with

the multiplication in C[N ] [BKK19]. Geometrically, this fusion product is viewed inside the Beilinson-

Drinfeld Grassmannian Gr0,A, which is a family Gr0,A → A1 such that the generic fibre is isomorphic to

Gr × Gr and the 0-fibre is isomorphic to Gr. Then to compute the fusion product of two MV cycles Z1

and Z2, we take the family of Gr0,A over A1 \ {0} whose fibres are isomorphic to Z1 × Z2. Taking the

closure of this, the top dimensional irreducible components of the 0-fibre constitute the terms in the

fusion product of Z1 ∗ Z2. We call this family Z1 ∗A Z2 and denote its 0-fibre by Z1 ∗0 Z2.

3

If we are able to find a stable MV cycle Z∗i such that

Zi ∗ Z∗i = Z+ + Z−

where Z+ and Z− corresponds to x+ and x−, respectively, then we can immediately conclude x∗i = bZ∗iHence, as all cluster variables are created through these exchange relations, we deduce that the cluster

variables are in the MV basis.

Exchange Relation for Polytopes and Cycles

In [GLS06], the authors gave a categorification of the cluster structure in C[N ] by associating the seed

Σ with a basic maximal rigid Λ-module T =⊕r

i=1 Ti along with the same matrix B, the xi with Ti, and

the exchange relations with non-split short exact sequences

0→ Ti → T+ → T ∗i → 0

0→ T ∗i → T− → Ti → 0.

Our first main result translates these short exact sequences in Λ-mod to an equation of MV polytopes

akin to the original exchange relations.

Theorem A. (Theorem 5.1.1) Pol(Ti) + Pol(T ∗i ) = Pol(T+) ∪ Pol(T−)

This equation allows us to compute Pol(T ∗i ) very easily by noting that

Pol(T ∗i ) = {x : x+ Pol(Ti) ⊂ Pol(T+) ∪ Pol(T−)}.

Theorem A is also a necessary condition for a corresponding exchange relation of MV cycles, as we

explain in Section 5.2. In [BKK19], the authors considered a Duistermaat-Heckmann measure DH on

MV cycles such that for an MV cycle Z, DH(Z) is supported on Pol(Z). Using this, it is possible to

show that by applying DH to a fusion product Z1 ∗Z2 = Z+ +Z−, we obtain the equation in Theorem

A. Hence the only possible candidate for Z∗i is the stable MV cycle whose MV polytope is Pol(T ∗i ). With

this candidate for Z∗i , we show in Chapter 6 the following result.

Proposition B. (Proposition 6.2.6) In type An for n ≤ 4, bZ∗i = ax∗i for some a ∈ Z≥1.

Extending Relationship Between Cycles and Modules

The proof of Proposition B relies on an interesting relationship between a stable MV cycle Z and its

corresponding generic module M . Each MV cycle has numerical invariants called valuations where for

each x ∈ C[N ], we obtain an integer valx(Z). Some of these invariants were used in [Kam10] to create

combinatorial data, called BZ data, that showed exactly when a polytope was an MV polytope. In

[BK10], the authors showed that if x was a cluster variable in an initial seed derived from a reduced

expression i and corresponds to the generic Λ-module X, then

valx(Z) = −dim Hom(X,M).

We extend this result to all cluster monomials x and modules M corresponding to a cluster monomial.

4

Theorem C. (Theorem 6.2.5) Let M be a generic module corresponding to a cluster monomial in C[N ]

and MV cycle Z. Let X be a module corresponding to a cluster monomial x. Then

valx(Z) = −dim Hom(X,M).

Calculating the Fusion Product

In Chapter 7, in joint work with Anne Dranowski and Joel Kamnitzer, we present a way to compute the

fusion product in type A. Normally, the product is computed by looking at intersection multiplicities

[BKK19], a very geometric point of view and difficult to calculate in general, but our approach computes

this product via matrices and semi-standard Young tableaux. We begin with showing a generalization

of the Mirkovic-Vybornov isomorphism [MV07b] [MV19].

Theorem D. (Theorem 7.4.6) We have an isomorphism

Oλ′,λ′′

0,A ∩ Uµ′,µ′′

0,A∼= Gr

λ′,λ′′

0,A ∩ Sµ′,µ′′

0,A .

Both sides of the isomorphism are families over A1 and the original Mirkovic-Vybornov isomorphism

is recovered when we take the 0-fibre of these families. The family Oλ′,λ′′

0,A denotes an orbit of matrices

where the fibre over s 6= 0 are matrices X conjugate to a two block matrix whose first block is the

Jordan block of size λ′ with eigenvalue 0 and whose second block is the Jordan block of size λ′′ with

eigenvalue s. The 0-fibre of this family consists of matrices conjugate to the Jordan block of size λ′+λ′′

with eigenvalue 0. The Mirkovic-Vybornov slice Uµ′,µ′′

0,A denotes a certain set of block upper triangular

matrices.

The generic fibre of Grλ′,λ′′

0,A ∩ Sµ′,µ′′

0,A is isomorphic to

(Grλ′

∩ Sµ′

− )× (Grλ′′

∩ Sµ′′

− )

while the 0-fibre is isomorphic to

Grλ′+λ′′ ∩ Sµ

′+µ′′

− .

When computing the fusion product of MV cycles Z1 and Z2, we have Z1 × Z2 ⊂ (Grλ′

∩ Sµ′

− ) ×(Grλ

′′

∩ Sµ′′

− ) for some λ′, λ′′, µ′, µ′′ and the terms in the fusion product are in Grλ′+λ′′ ∩ Sµ1+µ2

− so by

using Theorem D, we are able to view this family of MV cycles as a family of matrices instead.

Following [Dra20a], it is possible to find the ideal of the MV cycles appearing as a factor or as a

term in the fusion product through these matrices. These ideals can then be turned into a semi-standard

Young tableaux and each of these corresponds to an MV cycle [Dra20b]. Putting all these ideas together,

the steps involved in computing a fusion product of MV cycles Z1 ∗ Z2 are:

1. Create a generic matrix A in Uµ′,µ′′

0,A and use the fact that we also require A to be in Oλ′,λ′′

0,A to

create the ideal I for Z1 ∗A Z2

2. Find the ideal J for the 0-fibre Z1 ∗0 Z2 by adding s to I and deconstruct it into ideals Jα for each

irreducible component of Z1 ∗0 Z2

3. Translate each ideal Jα into a semi-standard Young tableau and find the corresponding MV cycle

Zα

5

4. We then conclude Z1 ∗ Z2 =∑α Zα.

We end with Chapter 8 by looking at the details of diagram (?) in type A3 and compute all fusion

products corresponding to an exchange relation through the method developed in Chapter 7. In par-

ticular, we show that the integer a in Proposition B is always 1, thus showing directly that the cluster

variables are in the MV basis up to type A3, culminating in our final result.

Proposition E. (Proposition 8.4.1) In type A3, every cluster variable in C[N ] is in the MV basis.

Chapter 1

MV Basis

Let G be a simple simply-connected complex algebraic group with Borel subgroup B, unipotent radical

N , and maximal torus T . In this chapter we consider the first class of objects we are interested in,

Mirkovc-Vilonen (MV) cycles, and see how they index a basis of C[N ].

1.1 MV Cycles

Let G∨ be the Langlands dual group of G with maximal torus T∨. Our choice of positive and negative

roots give dual opposite Borel subgroups B∨± with unipotent radicals N∨±. Let P denote the character

lattice of T and P+ the set of dominant weights. Note that P is also the cocharacter lattice of T∨.

Let R ⊂ P be the root lattice and R+ ⊂ R be the Z≥0-span of the positive roots. For λ, µ ∈ P , we

say λ ≥ µ if and only if λ− µ ∈ R+. Let {αi}i∈I denote the set of simple roots and {α∨i }i∈I denote the

set of simple coroots. Let ρ, respectively ρ∨, be the half-sum of the positive roots, respectively coroots.

Let O = C[[t]] be the ring of formal power series and K = C((t)) be its quotient field. The affine

Grassmannian Gr of G∨ is defined to be the quotient G∨(K)/G∨(O).

Given a cocharacter µ ∈ P , we obtain a map K× → T∨(K). Let tµ ∈ T∨(K) ⊂ G∨(K) denote the

image of t ∈ K× under this map. We will also use tµ to denote its image in Gr. These points are actually

the T∨ fixed points of Gr(C), that is,

(Gr(C))T∨

= {tλ : λ ∈ P+}.

Definition 1.1.1. For a dominant weight λ ∈ P+, the spherical Schubert cell is Grλ = G∨(O)tλ ⊂ Gr,the G∨(O)-orbit through tλ.

We call its closure Grλ a spherical Schubert variety and from [Zhu16, Proposition 2.1.5],

Grλ =⊔γ≤λ

Grγ .

Definition 1.1.2. For a weight µ ∈ P , the semi-infinite orbit is Sµ± = N∨±(K)tµ ⊂ Gr, the N∨±(K)-orbit

through tµ.

Theorem 1.1.1. [MV07a, Theorem 3.2a] For λ ∈ P+ and µ ∈ P , Grλ ∩ Sµ− is non-empty precisely when

tµ ∈ Grλ and in this case, Grλ ∩ Sµ− is pure of dimension 〈ρ∨, λ− µ〉.

6

Chapter 1. MV Basis 7

When Grλ ∩ Sµ− is non-empty, its irreducible components are called the Mirkovic-Vilonen (MV)

cycles of type λ and coweight µ. We let Z(λ)µ denote the set of all such cycles, and let

Z(λ) :=⊔µ∈P

Z(λ)µ

be the set of all MV cycles of type λ.

Proposition 1.1.2. [And03, Proposition 3] Let λ ∈ P+ and µ ∈ P . The irreducible components of

Grλ ∩ Sµ− are the irreducible components of Sλ+ ∩ Sµ− contained in Grλ.

We also have an action of P on Gr given by λ · [g] = [g]tλ which induces an isomorphism

S0+ ∩ S

µ−∼→ Sλ+ ∩ S

λ+µ− .

We will often be working with the irreducible components of S0+ ∩ S

µ−, which we shall call stable MV

cycles of coweight µ. We denote the set of such cycles as Z(∞)µ, and the set of all stable MV cycles

as

Z(∞) :=⊔

µ∈Q+

Z(∞)µ.

Thus by Proposition 1.1.2, we obtain a bijection

{Z ∈ Z(∞) : λ · Z ⊂ Grλ} ↔ Z(λ).

Example 1.1.3. This will be our running example thoughout the thesis. Let G = GL3. It is known

that Gr(1,0,0) ∩ S(0,1,0)− and Gr(1,1,0) ∩ S(1,0,1)

− are both irreducible and isomorphic to P1. We also know

Gr(2,1,0) ∩ S(1,1,1)− to have exactly two irreducible components, each isomorphic to P2.

1.2 MV Basis of C[N ]

For λ ∈ P+, let L(λ) be the irreducible representation of G with highest weight λ. We will now explain a

construction of a basis of C[N ] indexed by Z(∞) by first constructing bases for each L(λ) and gluing them

together. The basis of L(λ) comes from the following theorem, the Geometric Satake correspondence,

first proven in [Gin95] and later refined in [MV07a].

Theorem 1.2.1. [Gin95, Theorem 1.4.1][MV07a, Theorem 7.3] Let PG∨(O)(Gr) denote the category of

G∨(O)-equivariant perverse sheaves on Gr with C-coefficients and Rep(G) denote the category of finite-

dimensional C-representations of G. Then Rep(G) is equivalent to PG∨(O)(Gr) as tensor categories.

The proof of Theorem 1.2.1 used weight functors

Fµ := H〈2ρ,µ〉c (S+µ ,−) : PG∨O (Gr)→ VectC

where µ ∈ P . Let F =⊕

µ∈P Fµ. By [MV07a, Theorem 3.6], the fiber functor F is isomorphic to the

cohomology functor H∗(Gr,−). Using Tannakian formalism, the functor F fits inside a commutative


diagram

PG∨(O)(Gr) Rep(G)

VectC

∼=

F

where Rep(G)→ VectC is the forgetful functor.

For λ ∈ P+, let ICλ be the intersection cohomology sheaf of Grλ. These are the simple objects in

PG∨(O)(Gr) so under F , they correspond to L(λ). In fact, we have Fµ(ICλ) = L(λ)µ, the µ weight space

of L(λ).

Proposition 1.2.2. [MV07a, Proposition 3.10]

L(λ)µ ∼= Htop(Grλ ∩ Sµ−).

Since Grλ ∩ Sµ− is pure-dimensional, its irreducible components form a basis for L(λ)µ. Combining

all of the weight spaces together, we thus get a basis for L(λ) indexed by Z(λ), which we denote by

{[Z] : Z ∈ Z(λ)}.

In [BK07, Corollary 5.43], the authors define an embedding

Ψλ : L(λ)→ C[N ]

such that

C[N ] =⋃λ∈P+

Ψλ(L(λ)).

Proposition 1.2.3. [BKK19, Proposition 6.1] For each Z ∈ Z(∞), there exists a unique element

bZ ∈ C[N ] such that for any λ ∈ P+, we have

tλZ ⊂ Grλ ⇒ bZ = Ψλ([tλZ]).

We are thus able to glue the bases of the L(λ)’s to form a basis of C[N ] indexed by Z(∞), which we

call the MV basis.

Example 1.2.1. Let G = GL3. Let Z1 = Gr(1,0,0) ∩ S(0,1,0)− , Z2 = Gr(1,1,0) ∩ S(1,0,1)

− , and Z+, Z− be

the two MV cycles making up Gr(2,1,0) ∩ S(1,1,1)− (c.f. Example 1.1.3). Consider C[N ] = C[x12, x13, x23]

where the xij are coordinates on

N =

1 x12 x13

1 x23

1

: xij ∈ C

.

We have bZ1= x12, bZ2

= x23, and {bZ+, bZ−} = {x13, x12x23 − x13}. We shall take bZ+

= x13 and

bZ− = x12x23 − x13. Every element of the MV basis is of the form bαZ1bβZ+

bγZ− or bαZ2bβZ+

bγZ− for some

α, β, γ ∈ N.


1.3 Product of MV Cycles

We start with an alternative description of the affine Grassmannian. Let s ∈ C, Os = C[[t − s]], and

Ks = C((t − s)). Note that O = O0 and K = K0. As before, we can consider the affine Grassmannian

Grs = G∨(Ks)/G∨(Os). By [LS97, Propositions 3.8 and 3.10], we can also identify Grs with the set of

pairs (F , ν) where F is a G-bundle on A := A and ν is a trivialization of F over A\{s}.We can consider a more general ”two-point version” of Gr, where we fix one of the points to be 0.

Let

Gr0,A := {(F , ν, s) : F is a G-bundle on A, s ∈ A, ν a trivialization of F over A\{0, s}}

be the Beilinson-Drinfeld Grassmannian. We have a natural projection map π : Gr0,A → A. Let

U = A\{0}.

Lemma 1.3.1. [MV07a, Diagram 5.9] π−1(U) ∼= U × Gr × Gr and π−1(0) ∼= Gr.

We shall identify the fibre Gr0,s over s ∈ A\{0} with Gr0 × Grs. From [BGL20, Section 5.1], this

generic fibre can also be viewed as the quotient

G(C[t, t−1, (t− s)−1]/G(C[t]).

The following geometric viewpoint of the product of MV cycles is from [AK06, Section 3]. Take

Z1 ∈ Z(∞)−ν1 , Z2 ∈ Z(∞)−ν2 . Then for i = 1, 2, we have Zi ⊂ S0+ ∩ S

−νi− . For µ1, µ2 ∈ P , we

can consider a family Sµ1

± ∗A Sµ2

± in Gr0,A whose fibres over U are isomorphic to Sµ1

± × Sµ2

± and whose

fibre over 0 is isomorphic to Sµ1+µ2

± . We can define a family Z1 ∗A Z2 as the irreducible component of

S0+ ∗A S0

+ ∩ S−ν1− ∗A S−ν2− whose fibres over U are isomorphic to Z1 × Z2. The fibre of this family over

0, denoted Z1 ∗0 Z2, is a subset of S0+ ∩ S

−ν1−ν2− and its top dimensional components are MV cycles. If

these components are Zα for α in some indexing set, then we will denote the fusion product of Z1 and

Z2 as Z1 ∗ Z2 =∑α Zα.

The product of MV cycles Z1, Z2 corresponds to the product of bZ1, bZ2

in C[N ] [BKK19, Lemma

7.10]. For a precise description of the coefficients in this product we have the following result:

Theorem 1.3.2. [BKK19, Theorem 7.11] Let Z1 ∈ Z(∞)−ν1 , Z2 ∈ Z(∞)−ν2 . Then

bZ1bZ2 =∑Z

i(Z, π−1(0) · Z1 × Z2 × U)bZ

where the sum is over all Z ∈ Z(∞)−ν1−ν2 , and i(Z,D · V ) denotes the intersection multiplicity of Z in

D · V (see Section 7.5.2).

In general, the fusion product is difficult to compute. However, with joint work with Dranowski and

Kamnitzer [BDK21], we present a method of computing the product in type A via a generalization of

the Mirkovic-Vybornov isomorphism [MV07b][MV19] in Chapter 7.

Chapter 2

Semicanonical Basis

In this chapter, we define the second basis of C[N ] that we are concerned with. This was originally

constructed by Lusztig in [Lus00]. We now assume that G is a group of type ADE. Let n ⊂ g be the

associated Lie algebras of N ⊂ G. Let n be the rank of G and r be the number of positive roots of G.

Let U(n) be the universal enveloping algebra of n. The semicanonical basis is actually a basis for U(n),

but dualizing it gives a basis for U(n)∗ ∼= C[N ].

2.1 Preprojective Algebras

Let Q = (Q0, Q1, s, t) be a finite quiver of type ADE, where Q0 denotes the set of vertices of Q, Q1

the set of edges/arrows, and s, t : Q1 → Q0 the source and target maps. For an edge α ∈ Q1, we

will often draw it as s(α)α−→ t(α). Let Q = (Q0, H = Q1 t Q1, s, t) be the doubled quiver, where

Q1 = {α∗ : α ∈ Q1} with s(α∗) = t(α) and t(α∗) = s(α). Define ε : H → {±1} by

ε(α) =

1 if α ∈ Q1

−1 if α ∈ Q1

.

If α1 is a path in Q starting at i and ending at j, and α2 is a path in Q starting at j and ending at k,

we denote α2α1 as their concatenation, so a path starting at i and ending at k.

Definition 2.1.1. The preprojective algebra Λ is defined as the path algebra with relations

Λ := CQ/

∑α∈Q1

(α∗α− α∗α)

= CQ/

(∑α∈H

ε(α)αα∗

).

A Λ-module M is a pair ((Mi)i∈Q0 , (Mα)α∈H) where Mi is a vector space and Mα : Ms(α) →Mt(α) is

a linear morphism. We will let Λ-mod denote the category of finite dimensional modules over Λ. Hence

each simple, projective indecomposable, and injective indecomposable of Λ corresponds to a vertex in

Q0. Let Si denote the simple Λ-module at vertex i ∈ Q0 and Pi denote the projective(-injective)

indecomposable Λ-module at vertex i ∈ Q0. We define dimM =∑i∈Q0

(dimMi)αi.

Example 2.1.2. Suppose Q is an A2 quiver. Then modules for the preprojective algebra associated to Q

consist of representations of the quiver • •α

α∗subject to the relations αα∗ = 0 and α∗α = 0. This

10

Chapter 2. Semicanonical Basis 11

algebra has exactly 4 indecomposables: S1 = C 00

0, S2 = 0 C

0

0, P1 = C C

1

0, and

P2 = C C0

1. We then have dimS1 = α1, dimS2 = α2, and dimP1 = dimP2 = α1 + α2.

We will need a few notions and results concerning Λ later in the thesis.

Definition 2.1.3. For d = (di)i∈Q0, e = (ei)i∈Q0

∈ Z|Q0|, define a symmetric bilinear form (−,−) given

by

(d, e) = 2∑i∈Q0

diei −∑α∈Q1

(ds(α)et(α) + es(α)dt(α)).

Lemma 2.1.1. [CB00, Lemma 1] For Λ-modules X and Y we have

dim Ext1(X,Y ) = dim Hom(X,Y ) + dim Hom(Y,X)− (dim(X),dim(Y )).

In particular, dim Ext1(X,Y ) = dim Ext1(Y,X).

Theorem 2.1.2. [GLS07c, Theorem 3] For finite dimensional Λ-modules X and Y , there is a functorial

pairing

Ext1(X,Y )× Ext1(Y,X)→ C.

In other words, there exists an isomorphism φX,Y : Ext1(X,Y )∼→ DExt1(Y,X) such that for λ ∈

Hom(Y, Y ′), [η] ∈ Ext1(X,Y ), ρ ∈ Hom(X ′, X), [ε] ∈ Ext1(Y ′, X ′), we have

φX′,Y ′(λ ◦ [η] ◦ ρ)([ε]) = φX,Y ([η])(ρ ◦ [ε] ◦ λ).

For a short exact sequence η : 0→ Y → E → X → 0, λ ∈ Hom(Y, Y ′), ρ ∈ Hom(X ′, X), [η] ◦ ρ is the

short exact sequence obtained by pulling back E → X and ρ, while λ ◦ [η] is the short exact sequence

obtained by pushing out Y → E and λ. Note that we have (λ ◦ [η]) ◦ ρ = λ ◦ ([η] ◦ ρ).

The following result is considered classical. We will refer the reader to [DR92] and [GS03] for proofs.

Proposition 2.1.3. For the preprojective algebra Λ, the following hold:

1. Λ is of finite representation type if and only if the underlying graph of Q is Dynkin of type An for

n ≤ 4.

2. Λ is of tame representation type if and only if the underlying graph of Q is Dynkin of type A5 or

D4.

Remark 2.1.4. Λ has exactly 4 indecomposable modules in type A2, 12 in type A3, and 40 in type A4.

2.2 Semicanonical Basis

For a dimension vector ~d = (di)i∈Q0, let V~d =

⊕i∈I Vi be the graded vector space where dimVi = di.

We define Λ~d to be the variety of all possible Λ-module structures on V~d, which can be viewed as

Λ~d =

M ∈ Λ-mod : dimM =∑i∈Q0

diαi

,


all ~d-dimensional Λ-modules. This is acted on by G~d =∏i∈Q0

GLdi via conjugation. If M is a Λ-module,

we will let O(M) denote its orbit under the associated group action. Note that two modules are in the

same orbit if and only if they are isomorphic to each other.

Let M(Λ~d) denote the set of constructible functions on Λ~d and M(Λ~d)G~d denote those constructible

functions that are constant on the orbits of G~d. Let M =⊕

~dM(Λ~d)G~d . We can turn this into a

N|Q0|-graded associative C-algebra with an operation ∗ as follows. Let f1 ∈M(Λ ~d1), f2 ∈M(Λ ~d2

), and

let ~d = ~d1 + ~d2. For M ∈ Λ~d, let VM be the variety of all Λ-submodules of M of dimension ~d2. Define a

function φM,f1,f2 : VM → C by

φM,f1,f2(U) = f1(M/U)f2(U).

As in [Lus00, Section 2.1], define

(f1 ∗ f2)(M) =∑m∈C

mχ(φ−1M,f1,f2

(m) ∩ VM )

where χ is the Euler characteristic.

Example 2.2.1. Let O1 ⊂ Λ ~d1and O2 ⊂ Λ ~d2

be G ~d1- and G ~d2

-orbits. For M ∈ Λ~d, define

F(O1, O2,M) = {U ∈ VM : U ∈ O2,M/U ∈ O1}.

Then we have

(1O1∗ 1O2

)(M) = χ(F(O1, O2,M))

where 1X is the characteristic function on X.

Define M to be the subalgebra of M generated by the functions 1Λi where Λi denotes the variety

consisting of the single point Si. There is an algebra isomorphism U(n) → M given by sending ei to

1Λi . Let M~d =M∩M(Λ~d)G~d . This has a basis

{fZ : Z ∈ Irr(Λ~d)}

where Irr(Λ~d) denotes the irreducible components of Λ~d [Lus00, Theorem 2.7]. The function fZ is the

unique element of M~d that is equivalent to 1 on a dense open subset UZ of Z and 0 on a dense open

subset of all other irreducible components of Λ~d. Take UZ to be the maximal possible subset in Z on

which fZ equals 1. For some Z ∈ Irr(Λ~d) and M ∈ Z, we call the module M generic if M ∈ UZ .

The basis of U(n) corresponding to ⋃~d

{fZ : Z ∈ Irr(Λ~d)}

is called the semicanonical basis. We then get a dual basis of U(n)∗ ∼= C[N ] called the dual semi-

canonical basis. While this basis is indexed by irreducible components, we will often associate the basis

elements to a corresponding generic module.

A large class of generic modules are rigid modules. A module M is called rigid if Ext1Λ(M,M) = 0.

Proposition 2.2.1. [GLS06, Corollary 3.15] For a Λ-module M with dimension vector ~d, the following

are equivalent:

• The closure O(M) of O(M) is an irreducible component of Λ~d


• The orbit O(M) is open in Λ~d

• M is rigid

In particular, rigid modules are generic.

Example 2.2.2. Consider the same setup as in Example 2.1.2. The modules S1, S2, P1, and P2 are all

rigid, so also generic. In fact, we have Λ(1,1) = O(P1)tO(P2)t {S1 ⊕ S2}. The irreducible components

of Λ(1,1) are O(P1) = O(P1) t {S1 ⊕ S2} and O(P2) = O(P2) t {S1 ⊕ S2}. If we let bM ∈ C[N ] =

C[x12, x13, x23] denote the dual semicanonical basis element corresponding to a generic module M , then

bS1 = x12, bS2 = x23, bP1 = x12x23 − x13, and bP2 = x13. Each element of the dual semicanonical basis

is of the form bαS1bβP1

bγP2or bαS2

bβP1bγP2

for some α, β, γ ∈ N.

Chapter 3

Polytopes

In this chapter we look at some combinatorics associated to MV cycles and Λ-modules.

3.1 Polytopes from MV Cycles

Recall that the action of the torus T∨ acting on Gr have fixed points {tµ : µ ∈ P+}. For an MV cycle Z,

we can associate to Z the unique polytope

Pol(Z) := Conv{µ : tµ ∈ Z}

called its MV polytope. This polytope can also be defined in two different ways using certain combi-

natorial data.

3.1.1 BZ Datum

Let ω1, . . . , ωn denote the fundamental weights of G∨ and let W = N(T∨)/T∨ be the Weyl group of G∨,

generated by the simple reflections s1, . . . , sn. Let w0 ∈W be the longest word. Let I = {1, . . . , n}. For

each i ∈ I, let ψi : SL2 → G∨ denote the ith root subgroup of G∨. Define si = ψi

([0 1

−1 0

])∈ G∨

to be the lift of the simple reflection si. This allows us to define the lift of w = si1 · · · sik ∈ W as

w = si1 · · · sik . Let C = (aij) denote the Cartan matrix of G. Define

Γ =⋃

w∈W,i∈Iw · ωi

to be the chamber weights.

Let V be a vector space over C. Note that the space V ⊗K has a filtration

· · · ⊂ V ⊗ tO ⊂ V ⊗O ⊂ V ⊗ t−1O ⊂ · · · .

Define val : V ⊗ K → Z by val(u) = k if u ∈ V ⊗ tkO and u /∈ V ⊗ tk+1O. Choose a highest weight

vector vωi in the fundamental representation V (ωi) for G∨. For γ = w ·ωi ∈ Γ, let vγ = w ·ωi. Note that

as G∨ acts on V (ωi), G∨(K) acts on V (ωi)⊗K. For γ ∈ Γ, define constructible functions Dγ : Gr → Z

14

Chapter 3. Polytopes 15

by Dγ([g]) = val(g · vγ).

Given a collection of integers M• = {Mγ}γ∈Γ, consider the sets

A(M•) = {L ∈ Gr : Dγ(L) = Mγ for all γ ∈ Γ}

and

P (M•) = {x ∈ tR : 〈x, γ〉 ≥Mγ for all γ ∈ Γ}.

Definition 3.1.1. [Kam10, Section 3.3] The integers M• is called a BZ datum of weight (λ, µ) if

1. (Edge inequalities) For each w ∈W and i ∈ I,

Mwsi·ωi +Mw·ωi +∑j 6=i

ajiMw·ωj ≤ 0

2. (Tropical Plucker relations) For w ∈W , i, j ∈ I with wsi > w, wsj > w, i 6= j, and aij = aji = −1,

Mwsi·ωi +Mwsj ·ωj = min(Mw·ωi +Mwsisj ·ωj ,Mwsjsi·ωi +Mw·ωj )

3. Mωi = 〈λ, ωi〉 and Mw0·ωi = 〈µ,w0 · ωi〉 for all i.

Remark 3.1.2. The tropical Plucker relations for groups that are not simply-laced are more complicated

and can be found in [Kam10].

Theorem 3.1.1. [Kam10, Theorem 3.1] Let M• be a BZ datum of weight (λ, µ). Then A(M•) is an

MV cycle of type λ and weight µ, and each MV cycle arises this way for a unique BZ datum M•. The

associated MV polytope is equal to P (M•).

From [Kam10, Proposition 2.2], the vertices of P (M•) are {µw}w∈W which are given by

µw =∑i

Mw·ωiw · αi.

If M• is a BZ datum of weight (λ, µ), then the lowest vertex of P (M•) is µe = λ and the highest vertex

is µw0= µ.

If Z1 and Z2 are MV cycles in the same equivalence class, that is, Z1 = tλZ2 for some λ ∈ P , then

their polytopes differ by translation by λ. When working with stable MV cycles, we will use the polytope

whose lowest vertex is the origin, and we call such polytopes stable MV polytopes.

Example 3.1.3. Consider the MV cycles in Example 1.2.1. The chamber weights are

Γ = {ω1 = (1, 0, 0), ω2 = (1, 1, 0), s1ω1 = (0, 1, 0), s2ω2 = (1, 0, 1), s2s1ω1 = (0, 0, 1), s1s2ω2 = (0, 1, 1)}

and the BZ data for the stable MV cycles are

Mω1 Mω2 Ms1ω1 Ms2ω2 Ms2s1ω1 Ms1s2ω2

Z1 0 0 -1 0 0 -1

Z2 0 0 0 -1 -1 0

Z+ 0 0 -1 0 -1 -1

Z− 0 0 0 -1 -1 -1


This gives the stable MV polytopes

Pol(Z1) = Conv{0, α1} =

α1

0

Pol(Z2) = Conv{0, α2} =

α2

0

Pol(Z+) = Conv{0, α1, α1 + α2} = α1

0

α1 + α2

Pol(Z−) = Conv{0, α2, α1 + α2} = α2

0

α1 + α2

Moreover, every other stable MV polytope in type A2 is a Minkowski sum of Pol(Z1), Pol(Z+), and

Pol(Z−), or of Pol(Z2), Pol(Z+), and Pol(Z−).

3.1.2 Lusztig Datum

Let i = (i1, . . . , ir) be a reduced expression of w0. For each 1 ≤ k ≤ r, define wik = si1 · · · sik and wi

0 = e.

For 1 ≤ k ≤ r, define the positive root βik = wi

k−1αik . Every MV polytope P has a unique path in its

1-skeleton that goes through its vertices µe, µwi1, . . . , µwi

r= µw0

. Let n1, . . . , nr denote the lengths of

the edges in this path. We call the sequence n• = (n1, . . . , nr) the i-Lusztig datum for P . Note that

the vector between vertices µwik−1

and µwik

is a multiple, including zero multiple, of βik. Hence if our

MV polytope is stable, then the vertices in the distinguished path can be calculated using the i-Lusztig

datum: we have µe = 0 and for 1 ≤ k ≤ r,

µwik

=

k∑i=1

nkβik.

We can obtain the BZ datum for a stable MV polytope through its Lusztig datum and vice-versa as

well. We call a chamber weight γ ∈ Γ an i-chamber weight if γ = wik ·ωj for some k and j. Let Γi be the

set of all i-chamber weights and let γik = wik · ωik . From [Kam10, Section 4.3], the relationship between

the BZ datum M• and i-Lusztig datum n• for a stable MV polytope is that

Mγik

=∑l≤k

〈βik, γ

ik〉nl

and

nk = −Mwik−1·ωik

−Mwik·ωik

−∑j 6=ik

aj,ikMwik·ωj

.

Note that since we are dealing with a stable MV polytope here, we always have Mωi = 0 for all i ∈ I,

which is needed in calculating the i-Lusztig datum.

Although to calculate the entire BZ datum we would need to use different reduced expressions for w0,

we can obtain the associated MV cycle from just one reduced expression i. Take Mωi = 0 for all i ∈ Iand let Mγi

kbe the integers calculated from the i-Lusztig datum n•. Noting that Γi = {ωi}ni=1∪{γik}ri=1,

consider the set

Ai(n•) = {L ∈ Gr : Dγ(L) = Mγ for all γ ∈ Γi}.

Theorem 3.1.2. [Kam10, Theorem 4.2] For each i-Lusztig datum n•, Ai(n•) is an MV cycle and each


MV cycle arises this way for a unique i-Lusztig datum.

Example 3.1.4. Consider the setup in Example 3.1.3. For the reduced expression i = (1, 2, 1), we obtain

βi1 = α1, β

i2 = α1 + α2, β

i3 = α2.

Since each of these have length one, we see we have the following i-Lusztig data:

Pol(Z1) Pol(Z2) Pol(Z+) Pol(Z−)

n• (1,0,0) (0,0,1) (1,0,1) (0,1,0)

3.2 Polytopes from Λ-Modules

As in [BKT14, Section 1.3], we define the polytope of a Λ-module M to be

Pol(M) := Conv {dimN : N ⊂M is a submodule} .

This polytope was shown to be an MV polytope in [BK10].

Theorem 3.2.1. [BK10, Theorem 5] There exists a family of Λ-modules N(γ) indexed by weights γ ∈ Pcharacterized up to isomorphism by the following properties:

• If γ is antidominant, then N(γ) = 0.

• Let i ∈ I and γ be a W -conjugate of −ωi with γ 6= −ωi. Then N(γ) satisfies

dimN(γ) = γ + ωi and socN(γ) ∼= Si.

For a Λ-module M and a weight γ ∈ P , define Dγ(M) = − dim HomΛ(N(γ),M).

Theorem 3.2.2. [BK10, Proposition 12, Remark 14(i), Theorem 19] If M is a generic Λ-module, then

the collection of integers (Dγ(M))γ∈Γ is a BZ datum.

Then another formulation of Pol(M) is

Pol(M) = Conv {x ∈ tR : 〈x, γ〉 ≥ Dγ(M) for all γ ∈ Γ} ,

which we now know is a stable MV polytope by the previous theorem.

Example 3.2.1. Consider the situation in Example 2.1.2. The non-zero modules N(γ) are N(−s1ω1) =

S1, N(−s2ω2) = S2, N(−s2s1ω1) = P2 =: P+, and N(−s1s2ω2) = P1 =: P−. For each indecomposable

Λ-module, we have the Dγ values to be:

D−ω1D−ω2

D−s1ω1D−s2ω2

D−s2s1ω1D−s1s2ω2

S1 0 0 -1 0 0 -1

S2 0 0 0 -1 -1 0

P+ 0 0 -1 0 -1 -1

P− 0 0 0 -1 -1 -1

Comparing this with Example 3.1.3, we see that Pol(Z1) = Pol(S1), Pol(Z2) = Pol(S2), Pol(Z+) =

Pol(P+), and Pol(Z−) = Pol(P−).


Given an MV polytope, we can also construct a generic module using its i-Lusztig datum [BKT14,

Theorem 5.11]. For each i ∈ I, define an endofunctor Σi on Λ-mod as follows. For a Λ-module M =

((Mi), (Mα)), we have the following diagram at vertex i:

⊕α∈H,t(α)=i

Ms(α)(ε(α)Mα)−−−−−−→Mi

(Mα∗ )−−−−→⊕

α∈H,t(α)=i

Ms(α).

To simplify the notation, we shall rewrite this diagram as

Mi

Min(i)−−−−→Mi

Mout(i)−−−−−→ Mi

Note that the preprojective relation implies we have Min(i) ◦Mout(i) = 0, so imMout(i) ⊂ kerMin(i). Let

Mout(i) : Mi → kerMin(i) be induced by Mout(i). We then replace the above diagram with

Mi

Mout(i)Min(i)−−−−−−−−−→ kerMin(i) ↪→ Mi

to obtain the module ΣiM .

Given a reduced expression i = (i1, . . . , ir) of w0, by [BKT14, Proposition 7.4], we obtain a sequence

of indecomposable modules

Bi1 = Si1 , B

i2 = Σi1Si2 , . . . , B

ir = Σi1Σi2 · · ·Σir−1

Sir .

Note that we also have dimBik = βi

k for each 1 ≤ k ≤ r. For a generic Λ-module M , let n• = (n1, . . . , nr)

be the i-Lusztig datum for Pol(M). We then obtain a filtration

0 = F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fr = M

where Fi/Fi−1∼= (Bi

i)⊕ni . Moreover, up to isomorphism, there is a unique generic Λ-module with such

a filtration [BKT14, Theorem 4.4].

Example 3.2.2. Consider the situation in Example 3.1.4. With i = (1, 2, 1), we have

Bi1 = S1, B

i2 = P−, B

i3 = S2.

From the table in Example 3.1.4, the only one that gives a non-trivial filtration is (1, 0, 1), which

corresponds to a module of dimension α1 + α2 with S1 as a submodule. There are exactly two modules

that satisfy this: S1⊕S2 and P+. Of these, only P+ is generic, so the i-Lusztig datum (1, 0, 1) corresponds

to P+.

Chapter 4

Cluster Algebras

In this chapter we define cluster algebras and look at how they are related to the various objects we

defined earlier.

4.1 Background

Fix integers 1 ≤ m ≤ n. Let F be the field of rational functions over Q in n variables. A seed of F is

a pair Σ = (x, Q) where x = (x1, . . . , xn) is a free generating set of F and Q is a quiver with vertices

Q0 labelled by {1, . . . , n}. The variables x1, . . . , xm are called mutable and the variables xm+1, . . . , xn

are called frozen. We assume Q has no loops and no 2-cycles, and that there are no edges between the

vertices m+ 1, . . . , n. Note that instead of a quiver, we can use a n×m matrix B = (bij) where

bij = #{arrows i→ j} −#{arrows j → i}.

This matrix is called the exchange matrix.

For each 1 ≤ k ≤ m, we define a new seed µk(Σ) = (µk(x), µk(Q)) called the mutation of Σ in

direction k. We have µk(x) = (x1, . . . , x∗k, . . . , xn) where

x∗k =

∏i→k xi +

∏k→i xi

xk=

∏bik>0 xi +

∏bik<0

xi

xk.

These mutation equations are called exchange relations. The quiver µk(Q) is constructed from Q by

1. adding a new arrow i→ j for each pair of arrows i→ k and k → j;

2. reversing the orientation of every arrow with target or source k;

3. and erasing every pair of opposite arrows.

In terms of matrices, we obtain a new matrix µk(B) = (b′ij) where

b′ij =

−bij if i = k or j = k

bij + sgn(bik) max{0, bikbkj} otherwise

Note that µk(µk(Σ)) = Σ.

19

Chapter 4. Cluster Algebras 20

If we have a seed Σ = (x, Q), then x = {x1, . . . , xn} is called a cluster and its elements are called

cluster variables. The cluster monomials are the elements that are a product of cluster variables in

a fixed cluster. The cluster algebra is defined as the subalgebra generated by all cluster variables.

Fix a seed Σ = (x = {x1, . . . , xn}, B = (bij)) with x1, . . . , xm the mutable variables, xm+1, . . . , xn

the frozen variables, and let Q = (Q0, Q1, s, t) be the quiver corresponding to B. For 1 ≤ j ≤ m, define

polynomials

yj =∏i∈Q0

x−biji .

Proposition 4.1.1. [FZ07, Proposition 7.8] If B has full rank, then every cluster monomial z can be

written uniquely as

z = R(y1, . . . , ym)

n∏j=1

xgjj

where R is primitive, that is, R is a ratio of two polynomials, neither of which divisible by any yj. In

this case we define the g-vector of z to be gΣ(z) = (g1, . . . , gn).

The following result shows that these g-vectors determine cluster monomials and also obey a mutation

rule (note the sign differences due to a different convention).

Theorem 4.1.2. [DWZ10, Nag13] Let Σ = (x, B) be a seed. Then different cluster monomials have

different g-vectors. Furthermore, suppose z is a cluster monomial with gΣ(z) = (g1, . . . , gn). Assume Σ

is mutable at vertex k and that gµk(Σ)(z) = (g′1, . . . , g′n). Then

g′j =

−gk if j = k

gj + max(−Bjk, 0)gk +Bjk min(gk, 0) if j 6= k.

4.2 Cluster Structure for C[N ]

As before, let n be the rank of G and r be the number of positive roots. For each L(λ), choose a highest

weight vector vλ. In [BZ97], the authors defined for each fundamental weight ωi and for each w ∈ W ,

the generalized minor

∆ωi,w(ωi) = w · vωi .

If G = SLn+1(C), then the generalized minors are ordinary minors of a matrix. In this case, we

have ∆ωi,w(ωi) to be the minor whose rows, respectively columns, are labelled by {1, .., i}, respectively

{w(1), . . . , w(i)}.The following is a construction of an initial seed for C[B+ ∩ B−w0B−] [BFZ05]. Let Q be a quiver

for the Dynkin diagram corresponding to G. Let i = (i1, . . . , ir) be a reduced expression for w0 ∈ Wadapted to Q. This means that i1 is a source in Q, and for each 1 ≤ k ≤ r − 1, ik+1 is a source

in sik · · · si1(Q) where si(Q) is the quiver obtained from Q by reversing each arrow starting at i. Let

k ∈ {−n, . . . ,−1} ∪ {1, . . . , r} and let

v>k =

sirsir−1 · · · sik+1if k ≥ 1

w0 if k ≤ −1.


Set

∆(k, i) = ∆ω|ik|,v>k(ω|ik|)

where ik = −k if k ≤ −1. Define

k+ =

r + 1 if |i`| 6= |ik| for all ` > k

min{` : ` > k and |i`| = |ik|} otherwise.

We say k is i-exchangeable if k, k+ ∈ {1, .., r}. Let e(i) denote the set of all i-exchangeable elements.

Note that e(i) contains exactly r−n elements. Define a quiver Ai with vertices {−n, . . . ,−1}∪{1, . . . , r}.For two vertices k, ` in Ai such that k < ` and {k, `} ∩ e(i) 6= ∅, we have an arrow k → ` if k+ = `, and

an arrow `→ k if ` < k+ < `+ and a|ik|,|i`| = −1 where a|ik|,|i`| is the corresponding entry in the Cartan

matrix of G.

Theorem 4.2.1. [BFZ05, Theorem 2.10] We have{{∆(k, i) : k ∈ {−n, . . . ,−1} ∪ {1, . . . , r}}, Ai

}to be an initial seed for C[B+ ∩B−w0B−]. The mutable variables are those ∆(k, i) with k ∈ e(i).

In the case when G is simply-laced, the cluster algebra obtained is independent of the choice of the

reduced expression i [BFZ05, Remark 2.14]. In [GLS07a, Section 4.4], it is shown that if we remove the

variables in the initial seed corresponding to the n non-i-exchangeable elements in {1, . . . , r}, remove

the corresponding vertices in Ai, and restrict the generalized minors ∆(k, i) to N , then we obtain an

initial seed for C[N ], also independent of the choice of reduced expression i. In particular, each cluster

has r cluster variables, n of them being frozen.

Example 4.2.1. Let G = SL3 and i = (1, 2, 1). Then we have the following pieces of data:

k -2 -1 1 2 3

ik -2 -1 1 2 1

v>k w0 w0 s1s2 s1 e

∆(k, i)

∣∣∣∣∣x12 x13

x22 x23

∣∣∣∣∣ x13 x12

∣∣∣∣∣x11 x12

x21 x22

∣∣∣∣∣ x11

We have e(i) = {1} and the quiver Ai is

−1 2

1

−2 3

so the matrix Bi associated to Ai is Bi =

−1

1

0

1

−1

where the column is indexed by {1} and the rows are


indexed by {−2,−1, 1, 2, 3}. The non-exchangeable elements in {1, 2, 3} are {2, 3} so the initial seed for

C[N ] that we obtain is

Σ =

{∣∣∣∣∣x12 x13

1 x23

∣∣∣∣∣ , x13, x12,

},

−1

1

0

=

{x12, x13, x12x23 − x13} ,

0

1

−1

.

Note that this seed has only 1 exchangeable variable, x12, and the other variables are frozen. Mutating

in direction x12, we get the exchange relation

x∗12 =x13 + (x12x23 − x13)

x12= x23.

We then obtain the seed

µx12(Σ) =

{x23, x13, x12x23 − x13} ,

0

−1

1

and these are all the seeds in the cluster algebra. In fact, the seed µx12(Σ) would be the initial seed if

we instead took i = (2, 1, 2). Notice that each cluster monomial is of the form xα12xβ13(x12x23 − x13)γ or

xα23xβ13(x12x23 − x13)γ for some α, β, γ ∈ N. Comparing this with Examples 1.2.1 and 2.2.2, we see that

each cluster monomial is in both the MV basis and the dual semicanonical basis.

To determine the g-vectors with respect to Σ, we have the polynomial

y1 = x−113 (x12x23 − x13).

The g-vectors of the cluster variables in Σ correspond to the standard basis vectors, so

gΣ(x12) = (1, 0, 0); gΣ(x13) = (0, 1, 0); gΣ(x12x23 − x13) = (0, 0, 1).

We have

x23 = (y1 + 1)x−112 x13

so gΣ(x23) = (−1, 1, 0), which can also be calculated from the mutation rule for g-vectors, knowing that

gµx12 (Σ)(x23) = (1, 0, 0).

Remark 4.2.2. As we just saw, in type A2, there are exactly 4 cluster variables. There are 12 in type

A3, 40 in type A4, and infinitely many otherwise. This matches the number of indecomposables in Λ for

the corresponding type.

4.3 Cluster Structure for Λ-mod

We now give a categorification of the cluster structure of C[N ] via Λ-mod. This was constructed in

[GLS06] to show that the cluster variables in C[N ] are also in the dual semicanonical basis.

For a Λ-module M , define add(M) to be the full subcategory of Λ-mod consisting of all modules

isomorphic to direct summands of finite direct sums of copies of M . We say a rigid module T is maximal


if for every indecomposable T ′ such that T ⊕ T ′ is rigid, T ′ ∈ add(T ).

Let Σ(M) be the number of isomorphism classes of indecomposable direct summands of M .

Theorem 4.3.1. [GS05, Corollary 1.2] For every rigid Λ-module T , Σ(T ) ≤ r.

A rigid module T is called complete if Σ(T ) = r. Note that complete implies maximal.

An additive full subcategory T of Λ-mod is called maximal 1-orthogonal if for every module M ,

the following are equivalent:

• M ∈ T

• Ext1(M,T ) = 0 for all T ∈ T

• Ext1(T,M) = 0 for all T ∈ T .

We say a module T is maximal 1-orthogonal if add(T ) is maximal 1-orthogonal. Note that a maximal

1-orthogonal T is a generator-cogenerator of Λ-mod, that is, T has all indecomposable projective and

injective modules as direct summands. Since all projectives are also injectives in Λ, this is equivalent to

saying T has P1, . . . , Pn as direct summands.

Theorem 4.3.2. [GLS06, Theorem 2.2] Let T be a Λ-module. The following are equivalent:

• T is maximal rigid

• T is complete rigid

• T is maximal 1-orthogonal

Let T = T1⊕· · ·⊕Tr be a basic complete rigid Λ-module where each Ti is indecomposable. Without

loss of generality, Tr−n+1, . . . , Tr are projective. We can define a matrix CT = (cij) where

cij = dim HomΛ(Ti, Tj).

Let B(T ) = (bij) be the matrix obtained from −C−tT , the inverse transpose of −CT , after removing

the last n columns. For each 1 ≤ k ≤ r − n, it was shown in [GLS06, GLS12] that there exists an

indecomposable rigid Λ-module T ∗k with:

• dim Ext1Λ(Tk, T

∗k ) = dim Ext1

Λ(T ∗k , Tk) = 1,

• T ∗k ⊕ T/Tk is again a basic complete rigid Λ-module,

• and the non-split short exact sequences with end terms Tk, T∗k are of the form

0→ Tk →⊕bik>0

T biki → T ∗k → 0

0→ T ∗k →⊕bik<0

T biki → Tk → 0.

Define the mutation of T in direction Tk to be µTk(T ) = T/Tk ⊕ T ∗k .

Theorem 4.3.3. [GLS06, Theorem 2.6] For a basic complete rigid Λ-module T , 1 ≤ k ≤ r−n, we have

B(µTk(T )) = µk(B(T )).


We see that Λ-mod is a categorification of the cluster structure on C[N ], where the clusters correspond

to a basic maximal rigid module T , the cluster variables correspond to the indecomposable summands of

T , the frozen variables correspond to the indecomposable projectives, the cluster monomials correspond

to elements in add(T ), the exchange matrix is B(T ), and the exchange relations correspond to two non-

split short exact sequences. We will call the basic maximal rigid modules seeds, the indecomposable

summands of T cluster modules, and the non-split short exact sequences exchange sequences.

The initial seed for this cluster structure is constructed in [GLS07a] which we will present below.

Let Q = (Q0 = {1, . . . , n}, Q1, s, t) be a quiver for the Dynkin diagram corresponding to G. We form

another quiver ZQ = {ZQ0,ZQ1, s, t} where ZQ0 = Z×Q0, ZQ1 = Z×Q1, and for (i, α), (i, α∗) ∈ ZQ1,

we haves(i, α) = (i+ 1, s(α)) s(i, α∗) = (i, t(α))

t(i, α) = (i, t(α)) t(i, α∗) = (i, s(α)).

Let I ⊂ C[ZQ] be the ideal generated by the relations∑α∈Q1,s(α)=q

(i, α∗)(i, α)−∑

α∈Q1,t(α)=q

(i, α)(i+ 1, α∗)

for each (i, q) ∈ ZQ0, and let Λ = C[ZQ]/I, known as the universal, or Galois, cover of Λ. There

is a natural covering functor F : Λ → Λ given by (i, q) 7→ q which extends to a push-down functor

F : Λ-mod→ Λ-mod where for q ∈ Q0, (FM)q =⊕

i∈ZM(i,q).

We have a translation automorphism τ on Λ defined by τ(i, q) = (i+ 1, q) for (i, q) ∈ ZQ0. We also

have a Nakayama permutation ν on Λ defined by

ν(i, q) =

(i+ q − 1 + `(µ(q))− `(q), µ(q)) in type An

(i+ n− 2 + `(µ(q))− `(q), µ(q)) in type Dn

(p+ q + 2 + `(µ(q))− `(q), µ(q)) in type E6 and q ≤ 5

(p+ 5, µ(q)) in type E6 and q = 6

(p+ 8, µ(q)) in type E7

(p+ 14, µ(q)) in type E8

where `(q) is the number of arrows pointing towards 1 on the unique path from 1 to q in Q, and

µ(q) =

n+ 1− q in type An

2n− 1− q in type Dn with n odd and q ≥ n− 1

6− q in type E6 and q ≤ 5

q otherwise.

Define a function N : Q0 → N by the property that τN(q)(0, q) = ν(0, µ(q)). Let ΓQ be the full subquiver

of Λ which has vertices satisfying the equation

(i, q) = τ i−N(q)ν(0, µ(q))

where q ∈ Q0 and 0 ≤ i ≤ N(q). This is equivalent to taking the full subquiver between two copies of


Qop in ZQ obtained by the embeddings q 7→ (0, q) and q 7→ ν(0, q).

Remark 4.3.1. If we restrict the translation functor τ to ΓQ, then we obtain the Auslander—Reiten

quiver for CQ.

We have a total ordering

x(−n) < x(−n+ 1) < · · · < x(−1) < x(1) < · · · < x(n− r)

of the vertices in ΓQ defined by x(i) < x(j) if there are no paths in ΓQ from x(i) to x(j). Then

i = (F (x(−n)), . . . , F (x(−1)), F (x(1)), . . . , F (x(n−r))) will be a reduced expression for w0 ∈W adapted

to Q.

For each vertex (i, q) ∈ ΓQ, let I(i,q) be the injective indecomposable in CΓQ. Viewing CΓQ-mod ⊂ Λ-

mod we then have

T =⊕

(i,q)∈ΓQ

FI(i,q)

to be a basic maximal rigid Λ-module with the projective indecomposables to be {FI(0,q) : q ∈ Q0}. The

seed ({FI(i,q)}(i,q)∈ΓQ , B(T )) will correspond to the seed in C[N ] obtained from i where the correspon-

dence is ∆(k, i)↔ FIx(k) for each k ∈ {1, . . . , r} and ∆(k, i)↔ FIx(−n−k−1) for each k ∈ {−n, . . . ,−1}.

There is also an analogue for the g-vectors of cluster monomials. Let T = T1 ⊕ · · · ⊕ Tr be a basic

maximal rigid Λ-module. For a Λ-module M , define the r-vector rT (M) of M to be the vector whose

kth coordinate is

rT (M)k = dim Hom(Tk,M).

Then the g-vector of M with respect to T is defined as

gT (M) = rT (M)C−tT .

Proposition 4.3.4. [GLS12, Lemma 6.4] Let T be a basic maximal rigid Λ-module with exchange matrix

B(T ). Let Ti be an indecomposable summand of T that is not projective. Let M be a Λ-module with

g-vectors gT (M) = (g1, . . . , gr) and gµTi (T )(M) = (g′1, . . . , g′r). Then

g′j =

−gk if j = k

gj + max((C−tT )jk, 0)gk − (C−tT )jk min(gk, 0) if j 6= k.

It is easy to see that the g-vectors of the indecomposable summands of T correspond to the standard

basis vectors. Since the mutation for these g-vectors is the same as the mutation for g-vectors of cluster

variables in C[N ], we see that we also have a correspondence of g-vectors.

Example 4.3.2. Let G = SL3. We will take our A2 quiver Q to be the quiver 1 2a . The Galois

cover of Q is the quiver Λ =...


(1, 2)

(1, 1)

(0, 2)

(0, 1)

(−1, 2)

(−1, 1)

(1,a∗)

(1,a)

(0,a∗)

(0,a)

(−1,a∗)

...

with the relations that any composition of two arrows is 0. We then have ΓQ to be the subquiver

(1, 1)

(0, 2)

(0, 1)

(1,a)

(0,a∗)

The ordering of the vertices is (0, 1) < (0, 2) < (1, 1), so our reduced expression adapted to Q is

i = (F (0, 1), F (0, 2), F (1, 1)) = (1, 2, 1). The injective CΓQ-modules are:

I(0,1) =

0

C

C

0

1

, I(0,2) =

C

C

0

1

0

, I(1,1) =

C

0

0

0

0

.

Thus the basic maximal rigid Λ-module we get is

T = FI(1,1) ⊕ FI(0,1) ⊕ FI(0,2)

=

(C 0

0

0

)⊕(

C C0

1

)⊕(

C C1

0

)= S1 ⊕ P+ ⊕ P−.

The matrix CT isdim HomΛ(S1, S1) dim HomΛ(S1, P+) dim HomΛ(S1, P−)

dim HomΛ(P+, S1) dim HomΛ(P+, P+) dim HomΛ(P+, P−)

dim HomΛ(P−, S1) dim HomΛ(P−, P+) dim HomΛ(P−, P−)

=

1 1 0

0 1 1

1 1 1

.


Then

C−1T =

0 −1 1

1 1 −1

−1 0 1

Removing the last 2 columns of −C−tT , we have B(T ) =

0

1

−1

. We have S1 to be the only direction we

can mutate T in, and we see that the two non-split short exact sequences are

0→ S1 → P+ → S2 → 0

and

0→ S2 → P− → S1 → 0.

Hence µS1(T ) = S2⊕P+⊕P− and µS1(B(T )) =

0

−1

1

. Note that the seed (µS1(T ), µS1(B(T ))) is what

we would get if we instead used the quiver 1 2a and obtained the reduced expression i = (2,1,2).

The r-vectors are

rT (S1) = (1, 0, 1); rT (S2) = (0, 1, 0); rT (P+) = (1, 1, 1); rT (P−) = (0, 1, 1)

so the corresponding g-vectors after left multiplication by C−tT are

gT (S1) = (1, 0, 0); gT (S2) = (−1, 1, 0); gT (P+) = (0, 1, 0); gT (P−) = (0, 0, 1).

Comparing with Example 4.2.1, we see that the correspondence between the cluster variables of C[N ]

and the cluster modules in type A2 also preserves the entire cluster structure.

Remark 4.3.3. We see that all indecomposable modules in type A2 are also cluster modules. This is the

case in types A3 and A4 as well. In other types, we have infinitely many cluster modules, but not all

indecomposables will be cluster modules. We refer the reader to [GS05, Section 5] for an example of an

indecomposable A5 module that is not rigid, and hence cannot be a cluster module.

4.4 Lusztig Datum and g-vectors

Recall in Section 3.2 that for a reduced expression of w0, i = (i1, . . . , ir), and a generic Λ-module M , we

obtained a filtration for M where the successive quotients in the filtration were of the form (Bik)⊕nk . The

numbers nk form the i-Lusztig datum for Pol(M), and we will denote ni(M) = (n1, . . . , nr). Consider

the module V i = V i1 ⊕ · · · ⊕ V i

r where V ik is the unique module whose socle is Sik with dimension vector

ωik − si1 · · · sikωik . These summands are indecomposable and are also N(γ)’s. The module V i is known

to be a maximal rigid module [GLS11, Section 2.4]. The following result is due to Joel Kamnitzer.

Theorem 4.4.1. For every generic Λ-module M , we have

gVi

(M)k = ni(M)k − ni(M)k+


where we take ni(M)r+1 = 0.

Proof. The i-Lusztig datum ni(M) is linear in the BZ datum and part of the BZ datum is given by the

r-vector rVi

(M). Since the g-vector gVi

(M) is linear in the r-vector, we thus have the Lusztig datum

to also be linear in the g-vector. As this linear transformation is independent of the module M , we may

assume M = V ik for some k. In this case, it is known that

ni(V ik)j =

1 if j ≤ k and ij = ik

0 otherwise

Since gi(V ik) is the kth standard basis vector, we then have

ni(M)j =∑

k≥j,ik=ij

gVi

(M)k.

Inverting this equation, we conclude that gVi

(M)k = ni(M)k − ni(M)k+ as required.

We will let ϕi : Zr → Nr denote the map sending g-vectors for V i to i-Lusztig datum.

Example 4.4.1. Let i = (1, 2, 1). Then V i = S1 ⊕ P− ⊕ P+. Note that although V i is the same module

as T in Example 4.3.2, the order that we are labelling the summands is different. We have the following

information of the indecomposable Λ-modules:

S1 S2 P+ P−

ni (1,0,0) (0,0,1) (1,0,1) (0,1,0)

gVi

(1,0,0) (-1,0,1) (0,0,1) (0,1,0)

so we see that the linear transformation sending the i-Lusztig datum to the g-vector corresponding to

V i is given by the matrix

1 0 −1

0 1 0

0 0 1

.

There is also a relation between the Lusztig datum and mutations of g-vectors. We say the reduced

expressions i = (i1, . . . , ir) and i′ = (i′1, . . . , i′r) of w0 differ by a 2-move at position k if

ik = i′k+1, ik+1 = i′k, aik,ik+1= 0, and ij = i′j for all j 6= k, k + 1,

and they differ by a 3-move at position k if

ik−1 = ik+1 = i′k, ik = i′k−1 = i′k+1, aik−1,ik = −1, and ij = i′j for all j 6= k − 1, k, k + 1.

We say i and i′ differ by a braid move at position k if they differ by a 2-move or a 3-move at position k.

Proposition 4.4.2. [Kam10, Proposition 5.2] Let i and i′ be two reduced expressions for w0. Let P be

an MV polytope with i-Lusztig datum ni = (n1, . . . , nr) and i′-Lusztig datum ni′

= (n′1, . . . , n′r). Then

1. If i and i′ differ by a 2-move at position k, then n′k = nk+1, n′k+1 = nk, and n′j = nj for all

j 6= k, k + 1.

2. If i and i′ differ by a 3-move at position k, then for p = min{nk−1, nk+1}, n′k−1 = nk + nk+1 − p,

n′k = p, n′k+1 = nk−1 + nk − p, and n′j = nj for all j 6= k − 1, k, k + 1.


We will let µi′

i : Nr → Nr be the function that sends i-Lusztig datum to i′-Lusztig datum, assuming

the reduced expressions i and i′ differ by a single braid move.

Proposition 4.4.3. [GKS16, Proposition 5.12] Let i and i′ be two reduced expressions of w0 that differ

by a braid move at position k. Then for a Λ-module M , the following diagram commutes:

Zr Zr

Nr Nr

µk

ϕi ϕi′

µi′i

where the map µk is given by mutation of g-vectors in direction k along with a possible reordering.

The reason why a reordering is needed in the above proposition can be seen from the following

example. In type A2, if we let i = (1, 2, 1) and i′ = (2, 1, 2), then V i = S1 ⊕ P− ⊕ P+ and V i′ =

S2 ⊕ P+ ⊕ P−. However, µ1(V i) = S2 ⊕ P− ⊕ P+ so even though V i′ ∼= µ1(V i) as modules, the order of

their summands is different which leads to different g-vectors. Hence the relation between gVi

(M) and

gVi′

(M) is the mutation in direction k along with a possible reordering.

4.5 Summary

We have so far outlined the following diagram:

C[N ] {generic Λ-modules}

{stable MV cycles} {stable MV polytopes}

However, this diagram does not commute in general. It is known to commute in types An for n ≤ 3 and

conjectured for type A4. In types A5 and D4, there exists a stable MV cycle Z and generic Λ-module

M such that Pol(Z) = Pol(M), but their corresponding elements in C[N ] are different. The type A5

situation is discussed in [BKK19, Theorem 1.7]. In particular, this shows that the dual semicanonical

basis and the MV basis are different.

Nonetheless, the diagram does commute if we restrict to initial seeds. That is, if i is a reduced expres-

sion for w0, then by [BKK19, Remark 2.10, Theorem 5.2], each generalized minor ∆(k, i) corresponds

to an MV cycle with stable MV polytope the same as Pol(FIx(k)). Hence to show that the diagram

commutes for all cluster variables, equivalently that the cluster variables are in the MV basis, then it

suffices to find an analogue of the exchange relations for MV cycles and polytopes.

Chapter 5

Cluster Structure for MV Polytopes

In this section we show the analogue of the exchange sequences appearing in Section 4.3, and show that

they give a necessary condition if the cluster variables in C[N ] are also elements of the MV basis.

5.1 Exchange Relation coming from Λ-mod

Let T = T1 ⊕ · · · ⊕ Tr be a basic maximal rigid Λ-module with indecomposable Ti and assume that

Tr−n+1, . . . , Tr are projective. Recall that for 1 ≤ i ≤ r−n, dim Ext(Ti, T∗i ) = dim Ext(T ∗i , Ti) = 1. Let

the corresponding exchange sequences be

ε+ : 0→ Ti → T+ → T ∗i → 0

ε− : 0→ T ∗i → T− → Ti → 0.

This section is devoted to proving the following result, which can be thought of as an interpretation

of exchange relations for MV polytopes.

Theorem 5.1.1. We have

Pol(Ti) + Pol(T ∗i ) = Pol(T+) ∪ Pol(T−).

The following inclusion is straightforward and true for all short exact sequences.

Proposition 5.1.2. We have Pol(Ti) + Pol(T ∗i ) ⊃ Pol(T+) ∪ Pol(T−).

Proof. Let M ⊂ T+ be a submodule. Viewing Ti as a submodule of T+, consider the exact sequence

0→ Ti ∩Mι→M → coker ι→ 0.

As coker ι ∼= M/Ti∩M and T ∗i∼= T+/Ti, we have a well-defined injective morphism M/Ti∩M → T+/Ti

sending [m] 7→ [m]. Then coker ι is a submodule of T ∗i . Thus dimM = dimTi ∩M + dim coker ι ∈Pol(Ti) + Pol(T ∗i ) so Pol(T+) ⊂ Pol(Ti) + Pol(T ∗i ). The argument for Pol(T−) is similar.

The other inclusion will require more work and uses specific properties of the preprojective algebra

Λ.

30

Chapter 5. Cluster Structure for MV Polytopes 31

Definition 5.1.1. Let ε : 0 → T1 → T → T2 → 0 be a short exact sequence and A ⊂ T1, B ⊂ T2 be

submodules. We say (A,B) is a good pair for ε if there exists a short exact sequence 0 → A → M →B → 0 and morphism M → T making the following diagram commute:

0 A M B 0

0 T1 T T2 0

Lemma 5.1.3. Let ε : 0 → T1 → T → T2 → 0 be a short exact sequence and A ⊂ T1, B ⊂ T2 be

submodules. Suppose we have a short exact sequence 0 → A → M → B → 0. Then there exists a

morphism M → T causing (A,B) to be a good pair for ε if and only if the pushout S of A → M and

A→ T1 is equal to the pullback P of B → T2 and T → T2, as extensions of B by T1.

Proof. Suppose S = P as extensions of B by T1. Then we have the following commutative diagram:

0 A M B 0

0 T1 P B 0

0 T1 T T2 0

The composition M → P → T will make (A,B) a good pair for ε.

Now suppose we have a morphism M → T causing (A,B) to be a good pair for ε. We have the

following diagram:

0 A M B 0

0 T1 S B 0

0 T1 P B 0

0 T1 T T2 0

By the five lemma, it suffices to construct a morphism S → P that makes the diagram commute. Since

the morphism M → T makes the square

M B

T T2

commute and as P is the pullback of B → T2 and T → T2, we have a unique morphism M → P such

that the square

M B

P B


commutes. As A = ker(M → B) and T1 = ker(P → B), we have a commutative square

A M

T1 P

Since S is the pushout of A→M and A→ T1, we have a unique morphism S → P .

Lemma 5.1.4. Let ε : 0 → T1 → T → T2 → 0 be a short exact sequence and let A ⊂ T1, B ⊂ T2 be

submodules. Suppose (A,B) is a good pair for ε. Let A′, B′ be modules such that A ⊂ A′ ⊂ T1 and

B′ ⊂ B. Then (A′, B) and (A,B′) are good pairs for ε.

Proof. As (A,B) is a good pair, there exists a module M such that the following diagram commutes

and has exact rows:

0 A M B 0

0 T1 T T2 0

f

g

Then the exact sequence 0→ A→ f−1(B′)→ B′ → 0 will allow (A,B′) to be a good pair for ε.

By Lemma 5.1.3, as (A,B) is a good pair for ε and g−1(B) is the pullback of B → T2 and T → T2,

we have the following commutative diagram with exact rows and exact columns

0 0 0

0 A M B 0

0 T1 g−1(B) B 0

0 T1/A g−1(B)/M 0

0 0

Consider the following diagram with exact rows:

0 A T1 g−1(B)/M 0

0 A′ T1 coker ι 0ι

As coker ι ∼= T1/A′, we have a well-defined surjective morphism π : g−1(B)/M ∼= T1/A → T1/A

′ ∼=coker ι. Then coker ι ∼= (g−1(B)/M)/ kerπ. As kerπ is a submodule of g−1(B)/M , we have kerπ ∼=M ′/M for some module M ⊂ M ′ ⊂ g−1(B). Then coker ι ∼= g−1(B)/M ′ and M ′ will allow (A′, B) to

be a good pair for ε.

Proposition 5.1.5. Let A ⊂ Ti and B ⊂ T ∗i be submodules. Then (A,B) is a good pair for ε+ or (B,A)

is a good pair for ε−.


Proof. Without loss of generality, suppose (A,B) is not a good pair for ε+. Then by Lemma 5.1.4, (A, T ∗i )

is not a good pair for ε+. Then by Lemma 5.1.3, for every short exact sequence 0→ A→M → T ∗i → 0,

we have the pushout of A → M and A → Ti to not be equal to the pullback of T ∗i=→ T ∗i and T → T ∗i ,

which is T , as extensions of T ∗i by Ti. As dim Ext1Λ(T ∗i , Ti) = 1, this means the pushout is the split

extension for all such M . Let ι : A → T1 be the injection. Then by Theorem 2.1.2, we have for all

[µ] ∈ Ext1Λ(T ∗1 , A),

φT∗i ,Ti(ι ◦ [µ])([ε−]) = φT∗i ,A([µ])([ε−] ◦ ιA).

As ι ◦ [µ] is the pushout of µ along ι, we get the split extension so ι ◦ [µ] = 0 in Ext1Λ(T ∗1 , A). Then

φT∗i ,A([µ])([ε−] ◦ ι) = 0

for all [µ] so [ε−] ◦ ι = 0. This means the pullback of T− → Ti and A → Ti is the split extension in

Ext1Λ(A, T ∗i ), which is also the pushout of 0→ A and 0→ T ∗i . Thus (0, A) is a good pair for ε−. Then

by Lemma 5.1.4, (B,A) is a good pair for ε−.

Remark 5.1.2. The condition that dim Ext1Λ(Ti, T

∗i ) = 1 can be weakened to dim Ext1

Λ(T ∗i , Ti) 6= 0. The

proof will then show that if (A,B) is not a good pair for any non-split extension of T ∗i by Ti, then (B,A)

is a good pair for every non-split extension of Ti by T ∗i .

Combining Propositions 5.1.2 and 5.1.5 gives us the proof of Theorem 5.1.1.

Corollary 5.1.6. Pol(T+) ∪ Pol(T−) is a convex polytope.

Remark 5.1.3. It is possible for one of Pol(T+) or Pol(T−) to be equal to Pol(Ti)+Pol(T ∗i ). The smallest

example is in type A3, where Ti = C C C1 0

0 1, T ∗i = C C C

0 1

1 0, T+ = S1⊕S3⊕P2,

and T− = P1 ⊕ P3. In this case, Pol(T+) = Pol(Ti) + Pol(T ∗i ) and Pol(T−) ( Pol(T+).

Using Theorem 5.1.1, we can construct the MV polytopes for the rigid modules obtained from

mutation. For sets of vectors A and B in a vector space V , define

A−B := {x ∈ V : {x}+B ⊂ A}.

If we already know Pi + P ∗i = Q as an equality of polytopes, then P ∗i = Q− Pi [Sch13, Lemma 3.1.11].

Thus we can combinatorially construct the MV polytope of a module obtained from mutation as

Pol(T ∗i ) = (Pol(T+) ∪ Pol(T−))− Pol(Ti).

5.2 Exchange Relation Coming from MV Cycles

In the previous section, we showed that Theorem 5.1.1 follows from the exchange sequences in Λ-mod.

We will now explain why it also is a necessary condition for a corresponding exchange relation of MV

cycles, in other words, the following result.

Proposition 5.2.1. Suppose we have a fusion product of MV cycles

Z1 ∗ Z2 = Z+ + Z−.


Then

Pol(Z1) + Pol(Z2) = Pol(Z+) ∪ Pol(Z−).

The following is based on Sections 9 and 10 of [BKK19]. It is well-known that there exists a projective

embedding Gr → P(V ) where V is an infinite-dimensional vector space (see [BKK19, Section 4.3]). For

n ∈ Z, let O(n) denote the usual line bundle on P(V ). Recall that for an MV cycle Z ⊂ Gr, Z is a

T∨-invariant subvariety. Let [Γ(Z,O(n))] denote the class of the space of sections Γ(Z,O(n)) in R(T∨),

the complexified representation ring of T∨. We view R(T∨) inside the space of distributions of (t∨R)∗ = tR

using the map

[U ] 7→∑µ∈P∨

dimUµδµ.

Let τn : tR → tR be the automorphism corresponding to scaling by 1n .

Definition 5.2.1. For an MV cycle Z, define its Duistermaat-Heckmann measure DH(Z) as the weak-

limit

DH(Z) = limn→∞

1

ndimZ(τn)∗[Γ(Z,O(n))]

in the space of distributions of tR.

For each p ∈ ZT∨ , let ΦT∨(p) be the weight of the action of T∨ on the fibre of O(1) at p. Define

pol(Z) = Conv{ΦT∨(p) : p ∈ ZT∨} ⊂ tR.

Proposition 5.2.2. [BP90] The measure DH(Z) is well-defined and has support pol(Z).

In [BKK19, Section 4.3], they defined a map ι : P → tR, and after extending this to a linear bijection

ι : t∗R → tR, they showed that for every stable MV cycle Z, we have an equality ι(Pol(Z)) = pol(Z).

Hence we will think of the support of DH(Z) as Pol(Z).

Suppose we had an equation of MV cycles Z1 ∗ Z2 = Z+ + Z−. Geometrically, this means that

Z1 ◦Z2 = Z+∪Z−. Note that Definition 5.2.1 can be extended to the case where Z is a subscheme, as in

[BKK19, Section 9.1], so we can make sense of DH(Z1 ◦Z2) and DH(Z+ ∪Z−). By [BKK19, Theorem

9.8], we have

DH(Z+ ∪ Z−) = DH(Z+) +DH(Z−)

which is supported on Pol(Z1) ∪ Pol(Z2). From [BKK19, Theorem 10.2], as DH is an algebra map, we

conclude

DH(Z1 ◦ Z2) = DH(Z1) ∗DH(Z2)

where the product on the right is the convolution of measures, and this is supported on Pol(Z1)+Pol(Z2)

[BKK19, Section 8.3]. Combining these, we thus have

Pol(Z1) + Pol(Z2) = Pol(Z+) ∪ Pol(Z−).

Chapter 6

Investigating the Cluster Structure

for MV Cycles

In this chapter we look at the exchange relations from the point of view of MV cycles. We will do this

by relating the fusion product with short exact sequences in Λ-mod.

6.1 Valuations

Recall the definition of val in Section 3.1.1. We will use vals to denote the same function val, but

constructed using Os and Ks instead. We will look at how the valuation of terms in a fusion product

compares with the valuations of the factors of the product, but we will first need to recall the ind-

structure of Gr and Gr0,A. Since we are considering stable MV cycles Z ⊂ S0+ ∩ S

µ−, we will be focused

on the dense subset Z ∩ S0+ = Z ∩N∨(K)/N∨(O) of Z. Hence we will be only looking at the parts of

the affine Grassmannian and Beilinson-Drinfeld Grassmannian defined using N∨ instead of G∨.

Take a faithful representation ι : G∨ ↪→ GL(V ) for a suitable m-dimensional vector space V . The

ind-structure on G∨(K) is defined using functors (G∨)(n) : C-alg→ Set, n ∈ N, where

(G∨)(n)(R) = {A ∈ G(R((t))) : the matrix coefficients of A,A−1 have poles of order ≤ n}.

The ind-structure on N∨(K) is defined similarly using functors (N∨)(n) whose definition is analogous to

(G∨)(n).

Now restrict the faithful representation ι : N∨ ↪→ GL(V ) ⊂ End(V ). We will also use N∨ to denote

the image of N∨ in End(V ). As N∨ is unipotent, by [Bor91, Corollary 4.8], we can choose a basis for

V such that N∨ is mapped into a closed subgroup of U , the group of upper triangular matrices with 1

along the diagonal, which is closed in End(V ). Then by transitivity, N∨ is closed in End(V ).

From now on, assume the closed embedding ι : N∨ ↪→Mm, where Mm := Mm(C) is the set of m×mmatrices, factors through U . Let x ∈ C[N∨]. As ι is a closed embedding, we have ι∗ : C[Mm]→ C[N∨]

to be a surjection. Note that for x ∈ (ι∗)−1(x) and η ∈ N∨(Ks), we have

x(η) = ι∗(x)(η) = x(ι(η))

35

Chapter 6. Investigating the Cluster Structure for MV Cycles 36

so valsx(η) = valsx(ι(η)).

For s ∈ C, define

V sk = {g ∈ N∨(Ks) : valsx(g) ≥ k}.

Lemma 6.1.1. V sk is closed in N∨(Ks).

Proof. Let x ∈ C[N∨]. Viewing x as a polynomial in matrix entries, it is clear that there exists functions

ci ∈ (N∨)(n) such that for every A ∈ (N∨)(n)(R), we have

x(A) =∑

i≥−n deg x

ci(A)(t− s)i.

Then valsx(A) ≥ k if and only if ci(A) = 0 for −ndeg x ≤ i < k. These vanishing equations show that

V sk ∩ (N∨)(n) is closed in (N∨)(n) so V sk is closed in N∨(Ks).

In the proof above, if k is chosen so that k < −ndeg x, then this would give (N∨)(n) ⊂ V sk so each

stratum is contained in some V sk . Fix a stratum (N∨)(n) and let A = (aij) ∈ (N∨)(n)(R), B = (bij) ∈N∨(Os) = N (0)(R). Consider

AB = (∑t

aitbtj).

Since ait has poles of order at most n and btj has no poles, then aitbtj has poles of order at most n, so∑t aitbtj has poles of order at most n. Thus (N∨)(n) is stable under right multiplication by N∨(Os). This

implies that if Z∩S0+ ⊂ N∨(Ks)/N∨(Os) is dense in an MV cycle Z and πs : N∨(Ks)→ N∨(Ks)/N∨(Os)

is the quotient map, then π−1s (Z ∩ S0

+) is contained in some stratum (N∨)(n) so π−1s (Z ∩ S0

+) ⊂ V sk for

some k. Choose k maximal with respect to this property and define valsx(Z) = k.

From [BGL20, Section 5.1], instead of viewing the Beilinson-Drinfeld Grassmannian as G-bundles

with a trivialization, we can instead use

Gr0,A = {(s, [g]) : s ∈ C, [g] ∈ G∨(C[t, t−1, (t− s)−1])/G∨(C[t])}.

As we are focused on the part of the Grassmannian defined using N∨ instead of G∨, we will instead

have Gr0,A denote

{(s, [g]) : s ∈ C, [g] ∈ N∨(C[t, t−1, (t− s)−1])/N∨(C[t])}.

Let

Gr0,A = {(s, g) : s ∈ C, g ∈ N∨(C[t, t−1, (t− s)−1])}

and let q : Gr0,A → Gr0,A be the quotient map. Define for s ∈ C \ {0} and k ∈ Z,

V ◦k = {(s, g) ∈ Gr0,A : val0x(g) + valsx(g) ≥ k}

and let Vk be the closure of V ◦k in Gr0,A. We are interested in the 0-fibre of Vk, but first a technical

lemma, whose proof was partly communicated by Marko Riedel.

Lemma 6.1.2. Let p(t) =∑di=0 ait

i be a polynomial. Let k ≥ 0 be an integer. In the coordinate ring

C[a0, . . . , ad, s, s−1] of a polynomial and a non-zero scalar, consider the ideal

I = 〈a0, . . . , ak−1〉 ∩ 〈a0, . . . , ak−2, p(s)〉 ∩ · · · ∩ 〈p(s), . . . , p(k−1)〉


where p(j)(s) denotes the jth derivative of p evaluated at s. Let J = (I ∩ C[a0, . . . , ad, s]) + 〈s〉. Then

ai ∈√J for all i = 0, . . . , k − 1.

Proof. We see that q0 := −a0p(s) ∈ I so a20 ∈ J . Hence a0 ∈

√J . Define

q1 := s−1(a0p(1)(s)− a1p(s)) = 2a0a2 − a2

1 + s(...) ∈ I.

Then 2a0a2 − a21 ∈ J ⊂

√J so a1 ∈

√J . For n > 1, define

qn := s−n

n∑j=0

(−1)j+1

j!an−jp

(j)(s) +∑

n/2≤j≤n−1

(−1)n−j+1

(j

n− j

)s2j−nqj

.

It is easy to see that qn ∈ I by induction on n. Define

pn :=

d∑j=n

(n∑k=0

cj,kan−kaj+k

)sj−n

where

cj,k =

−1 if j = n, k = 0

2 if j = n, k = 1

0 if j < n or k < 0

cj−1,k − cj,k−1 otherwise.

We will show by induction on n that qn = pn and an ∈√J . Note first that for m ≥ 1, if we let

Amk = cm+n−k,k, then up to sign, the numbers Am0 , . . . , Amm constitute the mth row of the Lucas triangle

[Fei67]. In particular, we have

Amk = (−1)k+1m+ k

m

(m

k

).

Then if we define A00 = −1, we have

pn = A00a

2n + A1

1an−1an+1 + · · · + Anna0a2n

+ (A10anan+1 + A2

1an−1an+2 + · · · + An+1n a0a2n+1)s

...

+ (Ak0anan+k + Ak+11 an−1an+k+1 + · · · + Ak+n

n a0a2n+k)sk

...

It is easy to check that q0 = p0, q1 = p1, and we have shown a0, a1 ∈√J , so suppose qj = pj and aj ∈

√J

for each j = 0, . . . , n − 1. Fix j ≤ n2 and k < n. We will show that the coefficient of ajan−j+ks

k in

snqn is 0. This coefficient is comprised of three possible values: one acquired from ajp(n−j)(s), another

from an−j+kp(j−k)(s) if j ≥ k. and the third from qn/2+` for each 0 ≤ ` ≤ bk/2c if n is even, otherwise

q(n+1)/2+` for each 0 ≤ ` ≤ b(k−1)/2c if n is odd. Through induction, these coefficients are respectively

(−1)n−j+1

(n− j + k

k

), (−1)j−k+1

(j

k

),

(−1)n2 +`+1

(n2 +`n2−`)An2−j+k−`n2−j+`

if n is even

(−1)n+12 +`

(n+12 +`

2`+1

)An+12 −j+k−`

n+12 −j+`

if n is odd.


Suppose that n is even. Then

bk/2c∑`=0

(−1)n2 +`+1

(n2 + `n2 − `

)An2−j+k−`n2−j+`

=

bk/2c∑`=0

(−1)n−j(n

2 + `

2`

)n− 2j + kn2 − j + k − `

(n2 − j + k − `n2 − j + `

)

=

bk/2c∑`=0

(−1)j(n

2 + `

2`

)((n2 − j + k − `n2 − j + `

)+

(n2 − j + k − 1− `n2 − j − 1 + `

))

=

bk/2c∑`=0

(−1)j(n

2 + `

2`

)((n2 − j + k − `

k − 2`

)+

(n2 − j + k − 1− `

k − 2`

)).

Multiplying each coefficient by (−1)j , it suffices to show that

(n− j + k

k

)−bk/2c∑`=0

(n2 + `

2`

)((n2 − j + k − `

k − 2`

)+

(n2 − j + k − 1− `

k − 2`

))=

0 if j < k

(−1)k+1(jk

)if j ≥ k

.

The Egorychev method [Ego84] gives us the following relationship between binomial coefficients and

contour integrals: (n

k

)=

1

2πi

∫|z|=ε

(1 + z)n

zk+1dz.

Using this, we obtain

bk/2c∑`=0

(n2 + `

2`

)(n2 − j + k − `

k − 2`

)

=1

2πi

∫|z|=ε

(1 + z)n2−j+k

zk+1

bk/2c∑`=0

(n2 + `

2`

)z2`

(1 + z)`dz

=1

2πi

∫|z|=ε

(1 + z)n2−j+k

zk+1

1

2πi

∫|w|=γ

(1 + w)n2

w

∑`≥0

z2`(1 + w)`

(1 + z)`w2`dwdz

=1

2πi

∫|z|=ε

(1 + z)n2−j+k

zk+1

1

2πi

∫|w|=γ

(1 + w)n2

w

1

1− z2(1+w)w2(1+z)

dwdz

=1

2πi

∫|z|=ε

(1 + z)n2−j+k+1

zk+1

1

2πi

∫|w|=γ

(1 + w)n2

w

(w − z)(w(1 + z) + z)dwdz.

We need to choose a contour containing the simples poles w = z and w = −z1+z . Then we require∣∣∣∣ z

1 + z

∣∣∣∣ < γ.

As |z| < ε, this means ∣∣∣∣ z

1 + z

∣∣∣∣ < ε

1− ε

so it suffices to haveε

1− ε< γ.


For convergence of the geometric series, we also require∣∣∣∣z2(1 + w)

w2(1 + z)

∣∣∣∣ < 1,

which holds ifε2

1− ε<

γ2

1 + γ.

To have both ε1−ε < γ and ε( ε

1−ε ) <γ2

1+γ , it suffices to have

εγ <γ2

1 + γ.

Hence choose ε such that ε < γ1+γ . Then ε < γ so we also have the pole at w = z in this contour. For

the residue at the pole w = −z1+z , we have

1

2πi

∫|z|=ε

(1 + z)n2−j+k

zk+1

1

2πi

∫|w|=γ

(1 + w)n2

w

(w − z)(w + z/(1 + z))dwdz

=1

2πi

∫|z|=ε

(1 + z)n2−j+k

zk+1(1 + z)−

n2−z/(1 + z)

−z − z/(1 + z)dz

=1

2πi

∫|z|=ε

(1 + z)k−j

zk+1

1

z + 2dz.

Similarly, we also obtain the residue for

bk/2c∑`=0

(n2 + `

2`

)(n2 − j + k − 1− `

k − 2`

)

at w = −z1+z to be

1

2πi

∫|z|=ε

(1 + z)k−j−1

zk+1

1

z + 2dz.

Hence the residue at w = −z1+z for the entire sum is

1

2πi

∫|z|=ε

(1 + z)k−j−1(1 + z + 1)

zk+1

1

z + 2dz

=1

2πi

∫|z|=ε

(1 + z)k−j−1

zk+1dz =

(k − j − 1

k

).

For the residue at the pole w = z, we have

1

2πi

∫|z|=ε

(1 + z)n2−j+k+1

zk+1

1

2πi

∫|w|=γ

(1 + w)n2

w

(w − z)(w(1 + z) + z)dwdz

=1

2πi

∫|z|=ε

(1 + z)n2−j+k+1

zk+1(1 + z)

n2

z

z(1 + z) + zdz

=1

2πi

∫|z|=ε

(1 + z)n−j+k+1

zk+1

1

z + 2dz.


Similarly, for the other sum, the residue is

1

2πi

∫|z|=ε

(1 + z)n−j+k

zk+1

1

z + 2dz.

Adding these two residues, we have

1

2πi

∫|z|=ε

(1 + z)k−j−1(1 + z + 1)

zk+1

1

z + 2dz

=1

2πi

∫|z|=ε

(1 + z)k−j−1

zk+1dz =

(n− j + k

k

).

In summary,

(n− j + k

k

)−bk/2c∑`=0

(n2 + `

2`

)((n2 − j + k − `

k − 2`

)+

(n2 − j + k − 1− `

k − 2`

))=

(n− j + k

k

)−(k − j − 1

k

)−(n− j + k

k

)=−

(k − j − 1

k

)=−(k − j − 1)(k − j − 2) · · · (−j)

k!.

If j < k, one of the terms in the numerator is 0 so the entire term is 0. If j ≥ k, then all terms in the

numerator are negative and we obtain (−1)k+1(jk

)as required. The n being odd case is similar where

we require

(n− j + k

k

)−b(k−1)/2c∑

`=0

(n+12 + `

2`+ 1

)n− 2j + k

n−12 + k − `− j

(n−12 + k − `− jk − 2`− 1

)=

0 if j < k

(−1)k(jk

)if j ≥ k

.

Hence all coefficients of the sj terms in snqn with j < n are 0. In the alternating sum

n∑j=0

(−1)j+1

j!an−jp

(j)(s),

let Cj,k be the coefficient of an−kaj+ksj when j ≥ n. Then

Cj,k = (−1)k+1

(j + k

k

).

We see that

Cj−1,k − Cj,k−1 = (−1)k+1

(j − 1 + k

k

)− (−1)k

(j + k − 1

k − 1

)= (−1)k+1

((j + k − 1

k

)+

(j + k − 1

k − 1

))= (−1)k+1

(j + k

k

)= Cj,k,


which is the same recursion formula as pn. Since each of the qj for j ≤ n − 1 also satisfy the same

formula, so does qn. The term a2n only comes from −anp(s), so it has coefficient -1. The coefficient of

the term an−1an+1 comes from an−1p(1)(s), which is n+ 1, and qn−1, which is

(−1)n−(n−1)

(n− 1

n− (n− 1)

)= −(n− 1).

Hence the coefficient of an−1an+1 is 2. As these are the initial conditions of pn, we have shown that

qn = pn.

It is clear that pn = −a2n + p + sP where p is a polynomial where every term has at least one of

a0, . . . , an−1 as a factor and P is some polynomial. Then as pn = qn ∈ I, we have −a2n + p ∈ J ⊂

√J .

As a0, . . . , an−1 ∈√J , we thus have an ∈

√J .

Lemma 6.1.3. We have

Vk|s=0 = {(0, g) ∈ Gr0,A : val0x(g) ≥ k}.

Proof. We can define an ind-structure on Gr0,A using functors ˜(N∨)(n) : C-alg→ Set where

˜(N∨)(n)(R) ={(s,A) : s ∈ R,A ∈ N∨(R[t, t−1, (t− s)−1]), and

A,A−1 have the form∑

−n≤i,j≤n

Mijti(t− s)j where Mij ∈Mm(R)}.

Let fs = t(t − s). To see how ˜(N∨)(n) is affine, for a matrix A =∑−n≤i,j≤nAijt

i(t − s)j with

Aij ∈Mm, we can consider the matrix

fns A =∑i,j≥0

Aijti(t− s)j =

∑i≥0

Aiti

for some Ai ∈Mm.

Choose a lift of x in C[Mm], denoted by x. Since ι : N∨ →Mm is a closed embedding, then we can

instead consider val0,sx(A) := val0x(A) + valsx(A) for A ∈ ι(N∨)(C[t, t−1, (t − s)−1]) so consider the

set

V ◦k ={

(s,A) : s ∈ C \ {0}, A ∈ ι(N∨)(C[t, t−1, (t− s)−1]), val0,sx(A) ≥ k}.

Let (s,A) ∈ ˜(N∨)(n)(R) ∩ V ◦k and suppose is homogeneous of degree d. Then

x(fns A) = fdns x(A)

so

val0,sx(fns A) = val0,sfdns x(A) = 2dn+ val0,sx(A).

Suppose x is not homogeneous, but of degree d. Let M = (mij) ∈ Mm be an arbitrary matrix and

suppose that

x(mij) =

d∑i=0

Pi(mij)


where Pi(mij) is a polynomial of degree i. Let x′ ∈ C[Mm] be such that

x′(mij) =

d∑i=0

Pi(mij)md−i11 .

Note that x′ is homogeneous of degree d. As ι : N∨ ↪→Mr factors through U , we have the diagonal entries

of A = (aij) to all be equivalent to 1. In particular, a11 ≡ 1 so x(A) = x′(A). Hence val0,sx(A) ≥ k if and

only if val0,sx′(A) ≥ k. As x′ is homogeneous of degree d, this is equivalent to wanting val0,sx

′(fns A) ≥2dn+ k.

Let p(t) = x′(fns A). Then val0,sp ≥ 2dn+ k corresponds to the ideal

〈p(0), . . . , p(2dn+k−1)(0)〉 ∩ 〈p(0), . . . , p(2dn+k−2)(0), p(s)〉 ∩ · · · ∩ 〈p(s), . . . , p(2dn+k−1)(s)〉.

Hence by Lemma 6.1.2, the 0-fibre corresponds to the ideal

〈p(0), · · · , p(2dn+k−1)(0)〉

so we see that we obtain the set {(0, p) : val0p ≥ 2dn+ k}.

Proposition 6.1.4. Let Z1 and Z2 be MV cycles. We will view Z2 ⊂ Grs, for some s 6= 0, through the

isomorphism Gr ∼= Grs. Suppose

Z1 ∗ Z2 =∑α

Zα.

Then for each α and x ∈ C[N∨], we have

val0x(Zα) ≥ val0x(Z1) + valsx(Z2).

Proof. We can assume our MV cycles are stable, so take Z1∩S0+ ⊂ N∨(K)/N∨(O) and Z2∩N∨+(Ks)t0 ⊂

N∨(Ks)/N∨(Os). For p ∈ C, let πp : N∨(Kp) → N∨(Kp)/N∨(Op) be the quotient map. Suppose

π−10 (Z1 ∩ S0

+) ⊂ V 0k and π−1

s (Z2 ∩N∨+(Ks)t0) ⊂ V sl for some k and l. Then

q−1((Z1 ∗A1 Z2)|U ) ⊂ U × Vk × Vl

by using the isomorphism N∨(C[t, t−1(t − s)−1])/C[t] ∼= N∨(C[t, t−1])/C[t] × N∨(C[t, (t − s)−1])/C[t].

Since Vk+l is closed and

Vk+l|s6=0∼=

⋃a+b=k+l

V 0a × V sb ,

we have

q−1(Z1 ∗A1 Z2) ⊂ C∗ × V 0k × V sl ⊂ Vk+l

so q−1(Z1 ◦ Z2) ⊂ Vk+l|0 = V 0k+l. In particular, this means that for any MV cycle Z ⊂ Z1 ◦ Z2, we have

val0x(Z) ≥ val0x(Z1) + valsx(Z2).


6.2 Terms in the Fusion

We will determine the MV cycles in a fusion product by finding their associated generic Λ-modules. To

do this, we will translate the valuation inequality of Proposition 6.1 into a short exact sequence.

Let M and N be Λ-modules with dimM = dimN . Consider three partial orders on Λ-mod, the first

of which was created by [AdF85].

Definition 6.2.1. We say M ≤ext N if there exists modules Mi, Ui, Vi and short exact sequences

0→ Ui →Mi → Vi → 0

such that M = M1, Mi+1 = Ui ⊕ Vi, 1 ≤ i ≤ s, and N = Ms+1 for some s.

Definition 6.2.2. We say M degenerates to N , denoted M ≤deg N , if N ∈ O(M), the closure of the

orbit of M .

Definition 6.2.3. We say M ≤ N if [X,M ] := dim Hom(X,M) ≤ [X,N ] for all modules X.

We have M ≤ N if and only if [M,X] ≤ [N,X] for all modules X by [AR85]. In general, we have

M ≤ext N =⇒ M ≤deg N =⇒ M ≤ N (see [Bon96] for details), but if every indecomposable is rigid,

then the partial orders are equivalent to each other.

Theorem 6.2.1. [Zwa99, Theorem 2] Let A be an algebra. If every indecomposable of A is rigid, then

≤, ≤deg, and ≤ext coincide for all A-modules.

Let {x1, . . . , xr} be the initial cluster in C[N ] corresponding to a reduced expression i of w0 with

xr−n+1, . . . , xr being the frozen variables. Let Zi be the corresponding MV cycle [BKK19, Remark 2.10,

Theorem 5.2], so xi = bZi . Let Ti, be the corresponding rigid indecomposable module and T :=⊕r

i=1 Ti,

a basic maximal rigid module. Let xi be a mutable variable with mutation x∗i . Similarly, let T ∗i be the

mutation of Ti. Let Z∗i be the MV cycle whose MV polytope is Pol(T ∗i ). The question is if bZ∗i = x∗i .

From the exchange relation in C[N ], we have

xix∗i = x+ + x−

where x+ and x− are monomials in the xj , j 6= i. We also have nonsplit short exact sequences

0→ Ti → T+ → T ∗i → 0

0→ T ∗i → T− → Ti → 0

where T+ and T− are in add(T/Ti). In particular, T+ and T− are both rigid. We also have T± corre-

sponding to x±.

It suffices to show that bZibZ∗i = x+ + x−. Suppose bZibZ∗i =∑α bZα where Zα are MV cycles. Let

Mα be the generic Λ-module such that Pol(Mα) = Pol(Zα).

Lemma 6.2.2. If Mα ≤ext Ti ⊕ T ∗i , then Mα ∈ {T+, T−}.

Proof. By definition, there exist modules Mj , Uj , Vj and short exact sequences 0→ Uj →Mj → Vj → 0

such that Mα = M1,Mj+1 = Uj ⊕ Vj , 1 ≤ j ≤ s, and Ti⊕ T ∗i = Ms+1 for some s. In particular, we have

a short exact sequence of the form

0→ Ti →Ms → T ∗i → 0


or

0→ T ∗i →Ms → Ti → 0

As the only middle term in extensions of Ti by T ∗i or of T ∗i by Ti are T+ and T−, we have Ms ∈{T+, T−, Ti ⊕ T ∗i }.

Suppose Ms ∈ {T+, T−}, so in particular, Ms is rigid. Then as Us−1 ⊕ Vs−1 = Ms is rigid, the short

exact sequence

0→ Us−1 →Ms−1 → Vs−1 → 0

is split. Hence Ms−1 = Us−1 ⊕ Vs−1 = Ms. Repeating this argument, we get that Mj = Ms for all

j = 1, . . . , s. In particular, M1 = Mα ∈ {T+, T−}.

If Ms = Ti ⊕ T ∗i , then Ms−1 ∈ {T+, T−, Ti ⊕ T ∗i } and we repeat the argument. For the case where

Mα = M1 = Ti ⊕ T ∗i , note that Mα is generic as it comes from an MV cycle. Since T+ ≤ext Ti ⊕ T ∗iand T− ≤ext Ti ⊕ T ∗i , we have both rigid modules T+ and T− degenerate to Ti ⊕ T ∗i . This implies that

Ti⊕ T ∗i is an element of both irreducible components O(T+) and O(T−). If Ti⊕ T ∗i is also generic, then

Ti ⊕ T ∗i is in a unique irreducible component but O(T+) 6= O(T−) else O(T+) ∩O(T−) = ∅ is open and

dense. Hence Mα 6= Ti ⊕ T ∗i .

We now begin the process of showing [X,Mα] ≤ [X,Ti ⊕ T ∗i ] for all cluster modules X. Then in the

finite representation type case, as all indecomposables are cluster modules so also rigid, the previous

lemma will imply that the only terms that can show up in Zi ∗ Z∗i correspond to T+ or T−.

Lemma 6.2.3. Let x, x′ ∈ C[N ] be cluster monomials and Z an MV cycle. Then

val(x+ x′)(Z) = min{valx(Z), valx′(Z)}.

Proof. In [BZ01, Theorem 5.8], the authors used a Lusztig parametrization xi : (C×)r → N to place a

positive atlas on N so that if x is a cluster variable in an initial seed coming from a reduced expression i

and b ∈ Kr, then x(xi(b)) is a positive polynomial, that is, a polynomial with positive coefficients. Since

every cluster monomial is a rational function of positive polynomials in the cluster variables of an initial

seed, we have

val(x+ x′)(xi(b)) = min{valx(xi(b)), valx′(xi(b))}.

Instead of using the parametrization xi, we use the parametrization yi defined in [Kam10, Section 4.4].

Using [BZ01, Theorem 5.7], if xi(b) = yi(b′), then b′ is a rational function of positive polynomials in terms

of b, so x(yi(b′)) is a rational function of positive polynomials in terms of b′. If b′ = (b′1, · · · , b′r) ∈ Kr is

such that for each i, the leading coefficient of b′i is positive, then we are able to conclude that

val(x+ x′)(yi(b′)) = min{valx(yi(b

′)), valx′(yi(b′))}

as there will be no cancellations involving the leading coefficients. Using [Kam10, Theorem 4.2, Theorem

4.5], if val b′i = ni, then the image of yi(b′) in Gr is a generic point of the MV cycle Z ′ with i-Lusztig

datum n• and each MV cycle has a generic point of this form.

Let n• be the i-Lusztig datum of Z and choose b′ such that val b′ = ni and the leading coefficient of


each b′i is in R>0. Then we have

val(x+ x′)(Z) = val(x+ x′)(yi(b′))

= min{valx(yi(b′)), valx′(yi(b

′))}

= min{valx(Z), valx′(Z)}

as required.

Lemma 6.2.4. Let T =⊕Ti and R =

⊕Rk be basic maximal rigid modules and for each i, let

xi ∈ C[N ] be the cluster variable corresponding to Ti. Fix a mutable module Ti with exchange sequences

0→ Ti → T+ → T ∗i → 0 and 0→ T ∗i → T− → Ti → 0

and corresponding exchange relation xix∗i = x+ + x−. Let Z be an MV cycle with corresponding generic

module M such that M ∈ add(R). If [Ti,M ] = −valxi(Z) for all i, then [T ∗i ,M ] = −valx∗i (Z).

Proof. From [GLS07b, Section 7.3] and [GLS11, Proposition 12.4], we have [T+, R] 6= [T−, R] and for all

k,

[Ti ⊕ T ∗i , Rk] = max{[T+, Rk], [T−, Rk]}

so we obtain from [GLS11, Proposition 12.4(iii)] that for each k,

[Ti ⊕ T ∗i , Rk] =

[T+, Rk] if [T+, R] > [T−, R]

[T−, Rk] if [T+, R] < [T−, R].

In particular, since M ∈ add(R),

[Ti ⊕ T ∗i ,M ] = max{[T+,M ], [T−,M ]}.

Then by Lemma 6.2.3,

[T ∗i ,M ] = max{[T−,M ], [T+,M ]} − [Ti,M ]

= max{−valx+(Z),−valx−(Z)}+ valxi(Z)

= −min{valx+(Z), valx−(Z)}+ valxi(Z)

= −val(x+ + x−

xi

)(Z)

= −valx∗i (Z)

Theorem 6.2.5. Let M be a generic module corresponding to an MV cycle Z and assume that M

underlies a cluster monomial in C[N ]. Let X be a module corresponding to a cluster monomial x. Then

−valx(Z) = [X,M ].

Proof. Let (x1, . . . , xr) be an initial seed coming from a reduced expression i, so each xi is a generalized

minor. From the proof of [Kam10, Theorem 4.5] as well as the results in [BK10], for each i, there exists


γ ∈ Γ such that

valxi(Z) = Dγ(Z) = −[N(γ),M ] = −[Ti,M ].

Since M underlies a cluster monomial, M ∈ add(R) for some basic maximal rigid module R. Then

by Lemma 6.2.4, inducting on mutation, we have [X ′,M ] = −valx′(Z) for all cluster modules X ′ with

corresponding cluster variable x′. Let (x1, · · · , xr) be the cluster x is supported on and T = T1⊕· · ·⊕Trbe the basic maximal rigid module with X ∈ add(T ) such that xi corresponds to Ti. Then

X =

r⊕i=1

T⊕nii and x =

r∏i=1

xnii

so

valx(Z) = val

(∏i

xnii

)(Z) =

∑i

nivalxi(Z) = −∑i

ni[Ti,M ] = −[X,M ].

Proposition 6.2.6. In type An for n ≤ 4, we have bZ∗i = ax∗i for some integer a ≥ 1.

Proof. In the finite representation type case, it is known that a module is generic if and only if it is an

element of add(R) for some basic maximal rigid module R [CBS02, Theorem 1.1, Theorem 1.2] so using

Theorem 6.2.5 and Proposition 6.1.4, we obtain

[X,Mα] ≤ [X,Ti] + [X,T ∗i ] = [X,Ti ⊕ T ∗i ]

for all cluster modules X. Since all indecomposables are cluster modules in the finite representation type

case, we have Mα ≤ Ti⊕T ∗i . As every indecomposable is rigid, Theorem 6.2.1 implies Mα ≤ext Ti⊕T ∗i .

As Mα ∈ {T+, T−} by Lemma 6.2.2, we have

bZibZ∗i = ax+ + bx−

for some a, b ∈ Z≥0. Without loss of generality, suppose a ≤ b. Then

bZi(bZ∗i − ax∗i ) = (b− a)x−.

If a < b, then bZi = xi is a factor of x−, a contradiction [GLS13, Theorem 1.3]. Thus a = b so bZ∗i = ax∗ifor some a ≥ 1.

Chapter 7

Fusion Product in Type A

We show a method of computing fusion products in type A using a generalization of the Mirkovic-

Vybornov isomorphism [MV07b][MV19]. The following is also in [BDK21].

7.1 Notation

7.1.1 Rings and discs

Recall that for s ∈ C, Os := C[[t− s]] and Ks := C((t− s)), as well as O := O0, K := K0, O∞ := C[[t−1]],

and K∞ := C((t−1)). For any point s ∈ P = P1, Os is the completion of the local ring OP,s and thus

the formal spectrum of Os is the formal neighbourhood Ds of s, also known as the formal disc centered

at s. Similarly, the formal spectrum of the field Ks is the deleted formal neighbourhood, or punctured

disc, denoted D×s .

7.1.2 Groups

Let H be an algebraic group over C. We will be interested in H(R) where R is a C-algebra. Note

that evaluation at t = s provides a group homomorphism H(Os) → H. We denote the kernel of this

map by H1(Os), often called the first congruence subgroup. We will be particularly interested in this

construction in the case s =∞, which gives us the group H1(O∞).

Throughout this chapter, fix m ∈ N. We let G = GLm and we let T ⊂ G be the maximal torus of

diagonal matrices. We identify Zm with the coweight lattice of G and so a coweight ν = (ν1, . . . , νm) is

dominant if ν1 ≥ · · · ≥ νm and effective if νj ≥ 0 for all j. If ν is both effective and dominant, then

it is a partition of size |ν| = ν1 + · · · + νm ∈ N. Recall the positive root cone R+ ⊂ Zm. Explicitly, we

have

R+ = {(ν1, . . . , νm) : ν1 + · · ·+ νj ≥ 0 for j = 1, . . . ,m− 1 and ν1 + · · ·+ νm = 0} .

Given s ∈ C we define (t− s)ν to be the diagonal matrix

(t− s)ν1

(t− s)ν2. . .

(t− s)νm

47

Chapter 7. Fusion Product in Type A 48

which we can view in G(K) for any ring K containing (t− s)−1 and C[t]. For example, (t− s)ν ∈ G(K)

for K = Ks or C(t).

We will also be interested in the affine space Mm of m×m matrices. Note that for any C-algebra R,

G(R) consists of those matrices M ∈ Mm(R) whose determinant is invertible in R. Thus, for example,

(t− s)µ ∈Mm(C[t]) for all effective µ ∈ Zm but (t− s)µ ∈ G(C[t]) if and only if µ = 0.

7.1.3 Lattices

We will use the lattice model for the affine Grassmannian, so it is useful to recall the following definition.

Let R ⊂ K be two C-algebras. Consider Km as a K-module. By restriction, Km can be viewed as an

R-module. An R-lattice in Km is an R-submodule L ⊂ Km which is a free R-module of rank m and

satisfies L ⊗R K = Km. Equivalently, L = SpanR(v1, . . . , vm) where v1, . . . , vm are free generators of

Km.

We call Rm ⊂ Km the standard lattice. The group GLm(K) acts transitively on the set of R-lattices

in Km, thus giving a bijection between this set and GLm(K)/GLm(R), since GLm(R) is the stabilizer

of the standard lattice.

We will be particularly interested in C[t]-lattices in C(t)m. Given such a lattice L and a point a ∈ C,

the specialization of L at a is the lattice in Kma defined as L(a) := L⊗C[t]Oa. If L(a) = Oma then L is

said to be trivial at a. For example, the lattice (t− s)−1C[t] ⊂ C(t) is trivial at any a 6= s, since t− sis invertible in Oa.

7.2 Affine Grassmannians

We now redefine and define various versions of the affine Grassmannian. Each definition is made group-

theoretically and then restated as a moduli space of vector bundles and as a moduli space of lattices.

We also sketch how to pass between descriptions.

In these definitions, Vtriv denotes the trivial rank m vector bundle.

Recall we have the affine Grassmannian Grs = G(Ks)/G(Os) for s ∈ C with Gr = Gr0. It can also be

viewed as the moduli space of vector bundles with trivializations,

Grs ={

(V, ϕ) : V is a rank m vector bundle on Ds, ϕ : V∼−→ Vtriv on D×s

}.

As a moduli space of lattices,

Grs = {L ⊂ Kms : L is a Os-lattice} .

We obtain a lattice from a pair (V, ϕ) by setting L = Γ(Ds, V ) which is embedded into Kms = Γ(D×s , Vtriv)

using ϕ. On the other hand, to get a lattice from the group-theoretic description G(Ks)/G(Os), we set

L = gOms for g ∈ G(Ks).

Definition 7.2.1. The thick affine Grassmannian Gr = G(K∞)/G(C[t]).


Again, we have the following modular and lattice descriptions:

Gr ={

(V, ϕ) : V is a rank m vector bundle on P, ϕ : V∼−→ Vtriv on D∞

},

Gr = {L : L ⊂ Km∞ is a C[t]-lattice} .

Recall the two-point Beilinson–Drinfeld Grassmannian

π : Gr0,A → A

described in modular terms by

Gr0,A ={

(V, ϕ, s) : V is a rank m vector bundle on P, ϕ : V∼−→ Vtriv on P \ {0, s}

}.

The fibre Gr0,s over s is given by

G(C[t, t−1, (t− s)−1]/G(C[t]) .

We also have the lattice descriptions:

Gr0,A = {(L, s) : L ⊂ C(t)m is a C[t]-lattice trivial at any a 6= 0, s} ,

Gr0,s ={L : L ⊂ C[t, t−1, (t− s)−1]m is a C[t]-lattice

}.

Definition 7.2.2. The positive part of Gr, respectively Gr, is defined by

Gr+ = (Mm(O) ∩G(K)) /G(O) , respectively Gr+ = (Mm(C[t]) ∩G(K∞)) /G(C[t]) .

In modular terms, Gr+ (resp. Gr+) is the set of those (V, ϕ) where ϕ : V → Vtriv extends to an

inclusion of coherent sheaves over D0 (resp. over P). In lattice terms, Gr+ (resp. Gr+) contains those

lattices L which are contained in Om (resp. C[t]m).

7.2.1 Relations between different affine Grassmannians

These different versions of the affine Grassmannian are related as follows.

Proposition 7.2.1. There is a map Gr0,A → Gr defined in the following equivalent ways:

1. in modular terms as

(V, ϕ, s) 7→ (V, ϕ∣∣D∞

) ;

2. in the fibre over s ∈ A as the inclusion

G(C[t, t−1, (t− s)−1])/G(C[t])→ G(K∞)/G(C[t]) ;

3. in terms of lattices as

(L, s) 7→ L ,

using the inclusion C[t, t−1, (t− s)−1]m → Km∞ on ambient spaces.


The following result is the factorization property of the BD Grassmannian (see [Zhu16, Prop. 3.13]).

Proposition 7.2.2. The fibres Gr0,s of Gr0,A → A can be described as

Gr0,s∼=

Gr × Grs s 6= 0

Gr s = 0 .

In the modular realization, this isomorphism is given by restricting the vector bundle and trivialization.

Suppose that s 6= 0. In the lattice realization, this is given by forming the specializations

L 7→ (L(0), L(s)) .

Note that if L = gC[t]m for g ∈ G([t, t−1, (t− s)−1]), then

(L(0), L(s)) = (gOm, gOms ) .

In the s = 0 case, the isomorphism is described in the same way, except that we just need to form L(0).

We will let θs denote the isomorphism

θs : Gr0,s → Gr × Gr.

7.2.2 The fusion construction

We now consider a generalization of the fusion construction introduced in Section 1.3.

Let Gr0,A× denote the preimage of A× = A \ {0} under the map Gr0,A → A. The isomorphisms θs

glue together to a projection map

θ : Gr0,A× → Gr × Gr .

Let X1, X2 ⊂ Gr be two subschemes, and consider the subscheme

X1 ∗A× X2 := θ−1(X1 ×X2) ⊂ Gr0,A×

Given a point s ∈ A×, we write X1 ∗s X2 for the fibre of X1 ∗A× X2 over s. The map θs identifies

X1 ∗s X2 ⊂ Gr0,s with X1 ×X2 ⊂ Gr × Gr.We define X1 ∗A X2 to be the scheme-theoretic closure of X1 ∗A× X2 in Gr0,A. By construction, this

is a flat family over A. The fibre X1 ∗0 X2 of X1 ∗A X2 over 0 is a subscheme of Gr0,0, but regarded as

a subscheme of Gr using the isomorphism Gr0,0∼= Gr. We call this subscheme the (geometric) fusion of

X1 and X2.

7.2.3 Some subschemes of affine Grassmannians

Going forward, we fix a pair of arbitrary effective dominant coweights λ′, λ′′ of sizes N ′, N ′′ and a pair

of effective coweights µ′, µ′′ also of sizes N ′, N ′′ such that µ = µ′+µ′′ is dominant. Let λ = λ′+ λ′′ and

N = N ′ +N ′′.

Using λ, λ′, λ′′ and µ, µ′, µ′′ we define the subschemes of Gr, Gr, and Gr0,A which will be considered,

give some properties, and say how they are related.


Recall the spherical Schubert cell Grλ = G(O)tλ and semi-infinite orbits Sµ± = N±(K)tµ.

Definition 7.2.3. The family of two spherical Schubert varieties Grλ′,λ′′

0,A = Grλ′

∗A Grλ′′

.

By a theorem of Zhu, Grλ′

∗0 Grλ′′

is reduced and equal to Grλ ([Zhu16, Proposition 3.1.14]). If

s 6= 0, then the fibre Grλ′,λ′′

0,s contains the open locus Grλ′,λ′′

0,s = Grλ′ ∗s Grλ′′

a G(C[t])-orbit whose lattice

description is given in Lemma 7.4.2 below. Because we consider effective dominant λ′, λ′′, the spherical

Schubert varieties and their fusions lie in the positive part of the thick affine Grassmannian.

Lemma 7.2.3. The map Gr0,A → Gr restricts to a map Grλ′,λ′′

0,A → Gr+.

Proof. Let s 6= 0. Since λ′, λ′′ are effective, tλ′, (t − s)λ′′ ∈ Gr+. Since Gr+ is closed and invariant

under the action of G(C[t]), Grλ′,λ′′

0,s ⊂ Gr+.

For the s = 0 fibre, a similar reasoning applies (or we can simply conclude by taking closure).

Definition 7.2.4. The Kazhdan–Lusztig slice Wµ = G1(O∞)tµ ⊂ Gr.

In modular terms, Wµ corresponds to the locus of those (V, ϕ) such that V is isomorphic to the

trivial vector bundle on P and such that ϕ preserves the Harder–Narasimhan filtration of V at ∞. The

lattice description of Wµ ∩Gr+ is given in Lemma 7.4.3 below.

Definition 7.2.5. The family of two semi-infinite orbits Sµ′,µ′′

0,A ⊂ Gr0,A which we define to have fibres

Sµ′,µ′′

0,s := θ−1s (Sµ

′

− × Sµ′′

− ) for s 6= 0 and fibre θ−10 (Sµ) over s = 0.

We can also describe the fibres as orbits

Sµ′,µ′′

0,s = N−(C[t, t−1, (t− s)−1])tµ′(t− s)µ

′′⊂ G(C[t, t−1, (t− s)−1])/G(C[t]) .

See [BGL20, Section 5.2] for further details where this fibre is also described as the attracting locus for

a C× action.

In [KWWY14, Prop 2.6], it was observed that Sµ ⊂ Wµ when µ is dominant. More generally, we

have the following result.

Lemma 7.2.4. Under the map Gr0,A → Gr, the image of Sµ′,µ′′

0,A lands in Wµ.

Proof. Let L ∈ Gr be a lattice in the image of Sµ′,µ′′

0,A . So we can write L = gtµ′(t− s)µ′′C[t]m for some

g ∈ N−(C[t, t−1, (t− s)−1]).

Let h = (t − s)µ′′t−µ′′ . Note that h ∈ T1(O∞). Moreover, L = h(h−1gh)tµC[t]m. Since h−1gh ∈N−(K∞) we can factor h−1gh as n1n2 for some n1 ∈ (N−)1(O∞) and n2 ∈ N−(C[t]).

As µ is dominant, we see that t−µn2tµ ∈ N−(C[t]), and so L = hn1t

µC[t]m. Since hn1 ∈ G1(O∞),

the result follows.

7.3 Matrices

We now consider some subvarieties of the space of N ×N matrices, again using the coweights λ, λ′, λ′′,

µ, µ′, µ′′ fixed in the previous section.


7.3.1 Adjoint orbits and their deformations

Recall that λ = (λ1, . . . , λm) is a partition of N . Given a point s ∈ A we write Js,λ for the Jordan form

matrix with eigenvalue s and Jordan blocks of sizes λ1, . . . , λm.

Definition 7.3.1. The nilpotent (adjoint) orbit Oλ ⊂ MN (C) of matrices conjugate to J0,λ. These

matrices and the linear operators they represent will be said to have Jordan type λ.

Definition 7.3.2. For s ∈ A×, the (adjoint) orbit Oλ′,λ′′

0,s of matrices conjugate to J0,λ′ ⊕ Js,λ′′ . These

matrices and the linear operators they represent will be said to have Jordan type ((0, λ′), (s, λ′′)).

We recall that these orbits have closures which are given by rank conditions. More precisely, we have

Oλ = {A ∈MN (C) : rankAc ≤ N −# boxes in first c columns of λ, for c ∈ N}

and

Oλ′,λ′′

0,s = {A ∈MN (C) : rankAc ≤ N −# boxes in first c columns of λ′, for c ∈ N, and

rank(A− s)c ≤ N −# boxes in first c columns of λ′′, for c ∈ N} .(7.3.1)

Proposition 7.3.1. There exists a flat family Oλ′,λ′′

0,A → A whose fibre over s ∈ A is reduced and given

by Oλ′,λ′′

0,s if s 6= 0 and Oλ if s = 0.

In order to prove this proposition, we will need to recall some results from Eisenbud–Saltman [ES89].

Let r be a decreasing, non-negative function of N with r(0) = N , which we call a rank function.

Let k be maximal such that r(k) 6= 0. Let W be a subspace of Ak. Eisenbud and Saltman define and

study the flat family Xr,W →W whose fibres are reduced and defined by rank conditions. We will now

describe these fibres.

Let z ∈ Ak be a point. Some of the coordinates of z may be equal, so we introduce the following

data to keep track of these equalities. Let ` denote the number of distinct coordinates of z. There exist

a sequence of integers k = (k0, . . . , k`) such that 0 = k0 < k1 < · · · < k` = k, a point s ∈ A`, such that

sa 6= sb for any a 6= b, and a permutation p of {1, . . . , k} such that

sa = zp(ka−1+1) = zp(ka−1+2) = · · · = zp(ka)

for each a = 1, . . . , `. Given the choice of s we require that p is of minimal length. Given z, the data

k, s, p is unique up to a permutation of {1, . . . , `}.Next for a = 1, . . . , ` and c = 1, . . . , ka+1 − ka, we define

rz,a(c) := N +

c∑d=1

r(p(ka + d))− r(p(ka + d)− 1) .

By [ES89, Corollary 2.2], the fibre of the flat family Xr,W over z is given by

Xr,z = {A ∈MN (C) : rank(A− sa)c ≤ rz,a(c) for all c, a as above}. (7.3.2)

We will now apply these ideas to prove our Proposition 7.3.1.


Proof. For c ∈ N, let

r(c) = N −# boxes in first c columns of λ .

It is easy to see that this is a rank function, with k = λ1, the number of columns of λ.

Since λ = λ′+λ′′, there exists a permutation p of {1, . . . , k} (the columns of λ) such that p(1), . . . , p(k1)

are the columns of λ′ and p(k1 +1), . . . , p(k) are the columns of λ′′. Here k1 = λ′1 the number of columns

of λ′.

For s ∈ A, we define z(s) ∈ Ak byz(s)p(c) = 0 c = 1, . . . , k1

z(s)p(c) = s c = k1 + 1, . . . , k .

For s 6= 0, we see that the equalities in z(s) give rise to the data of ` = 2, k = (0, λ′1), s = (0, s) and

the permutation p. We also see that

rz(s),1(c) = N −# boxes in first c columns of λ′

rz(s),2(c) = N −# boxes in first c columns of λ′′ .(7.3.3)

On the other hand, for z(0), we get that ` = 1 and that

rz(0),1(c) = r(c) = N −# boxes in first c columns of λ . (7.3.4)

Let W = {z(s) : s ∈ C}. Combining Equations (7.3.1) to (7.3.4), we see that the Eisenbud–Saltman

family Xr,W →W gives our family Oλ′,λ′′

0,A .

7.3.2 The Mirkovic–Vybornov slice

Recall that µ = (µ1, . . . , µm) is a partition of N . The Mirkovic–Vybornov slice Tµ is the affine space

of N × N matrices of the form A = J0,µ + X where X is a µ × µ block matrix with possibly nonzero

entries A1ij , . . . , A

min(µi,µj)ij in the first min(µi, µj) columns of the last row of each µi × µj block.

By example, if µ = (3, 2), then A ∈ Tµ looks like

0 1 0 0 0

0 0 1 0 0

A111 A

211 A

311 A1

12 A212

0 0 0 0 1

A121 A

221 0 A1

22 A222

for some Akij ∈ C.

To A ∈ Tµ we will associate the m×m matrix of polynomials g(A) in Mm(C[t]) whose (i, j)th entry

is defined as follows.

g(A)ij =

tµi −∑µik=1A

kjit

k−1 i = j ,

−∑µik=1A

kjit

k−1 i 6= j .(7.3.5)


Continuing with the µ = (3, 2) example,

g(A) =

[t3 −A3

11t2 −A2

11t−A111 −A2

21t−A121

−A212t−A1

12 t2 −A222t−A1

22

].

In other words, the µi × µj block of A is used to produce a polynomial which is inserted in the (j, i)

entry of the m×m matrix g(A).

In Tµ, we will be interested in a certain family of block upper-triangular matrices.

Definition 7.3.3. The upper-triangular Mirkovic–Vybornov slice Uµ′,µ′′

0,A → A is defined by

Uµ′,µ′′

0,A := {(A, s) ∈ Tµ × A : g(A)ii = tµ′i(t− s)µ

′′i , g(A)ij = 0 for j < i } .

So a matrix in Uµ′,µ′′

0,A is weakly block upper-triangular and its diagonal blocks are given by the

companion matrices for the polynomials tµ′i(t− s)µ′′i where i = 1, . . . ,m.

For example, elements of U(1,1,0),(2,1,1)0,s look like

0 1 0 0 0 0

0 0 1 0 0 0

0 −s2 2s A112 A2

12 A113

0 0 0 0 1 0

0 0 0 0 s A123

0 0 0 0 0 s

.

Note that the fibre Uµ′,µ′′

0,0 is the same as the intersection Tµ ∩ n, where n denotes the set of strictly

upper-triangular matrices.

7.4 Mirkovic–Vybornov Generalization

Throughout this section, we continue our notation of the previous sections. So λ, λ′, λ′′ denote dominant

effective coweights, with λ′ + λ′′ = λ. Also µ, µ′, µ′′ are effective coweights with µ = µ′ + µ′′ and we

assume that µ is dominant.

7.4.1 Linear Operators from lattices

Lemma 7.4.1. Let V be a C[t]-module which is finite-dimensional as a complex vector space. For any

a ∈ A, the map

V → V ⊗C[t] Oa

restricts to an isomorphism between the generalized a-eigenspace of t and V ⊗C[t] Oa

Proof. For any b ∈ A, let Eb denote the generalized b-eigenspace of t. Then V = ⊕b∈CEb. Since t− b is

invertible in Oa and t− b acts nilpotently on Eb, we see that Eb ⊗C[t] Oa = 0.


So it suffices to show that Ea → Ea⊗C[t]Oa is an isomorphism. By the classification of modules over

C[t], it suffices to check this when Ea = C[t]/(t− a)k, where it is clearly true.

Lemma 7.4.2. Let L ∈ Gr+. Let s ∈ A×. The following are equivalent:

(i) L is in the image of the map Grλ′,λ′′

0,s → Gr+.

(ii) The linear operator t on C[t]m/L has Jordan type ((0, λ′), (s, λ′′)).

(iii) L ∈ G(C[t])tλ′(t− s)λ′′ .

Proof. First we recall that for L ∈ Gr+, L ∈ Grλ = G(O)tλ if and only if t|Om/L has Jordan type λ.

Now assume that (L, s) ∈ Grλ′,λ′′

0,s . By definition, L(0) ∈ Grλ′ and L(s) ∈ Grλ′′ . This means that t

acting on Om/L(0) has Jordan type λ′ and t acting on Oms /L(s) has Jordan type λ′′. For a = 0, s, we

see that

C[t]m/L⊗C[t] Oa ∼= Oma /L(a).

Lemma 7.4.1 shows that the map C[t]m/L→ Om/L(0) induces an isomorphism between the 0-generalized

eigenspace of t and Om/L(0). The same thing holds for the s-generalized eigenspace of t and Oms /L(s).

This shows that item (i) implies item (ii) and the logic can be reversed to see that item (ii) implies

item (i).

On the other hand, if L = gtλ′(t − s)λ

′′C[t]m for some g ∈ G(C[t]), then L(0) = gtλ′Om since

(t − s)λ′′ ∈ G(O). In this way, we see that item (iii) implies item (ii) and the logic can be reversed to

get equivalence.

7.4.2 Matrices from lattices

Lemma 7.4.3. Let L ∈ Gr+. The following are equivalent:

(i) L ∈ Wµ.

(ii) L = SpanC[t](v1, . . . , vm) for some vi of the form vi = tµiei +∑mj=1 pijej where pij ∈ C[t] has

degree less than min(µi, µj).

(iii) For all i,

tµiei ∈ SpanC({tkej : 0 ≤ k < min(µi, µj), 1 ≤ j ≤ m}) + L.

Moreover, for such L, βµ := {[tkei] : 0 ≤ k < µi, 1 ≤ i ≤ m} forms a basis for C[t]m/L.

Proof. Let L ∈ Wµ. Then L = SpanC[t](v1, . . . , vm) for some vi with vi = tµiei +∑j=1 qijt

µiej and

qij ∈ t−1O∞. Since L ∈ Gr+, we see that vi ∈ C[t]m which means that pij := qijtµi lies in C[t]. By

construction, the polynomial pij has degree less than µi.

Fix i and suppose that for some j, µj < µi. In this case, we can alter our basis to v′i = vi − qvj for

some polynomial q ∈ C[t]. This gives us new polynomials p′ij = pij − q(tµj + pjj). In this way, we can

ensure that pij has degree less than min(µi, µj). Thus item (i) implies item (ii).

Suppose that L = SpanC[t](v1, . . . , vm) as in item (ii). Then

tµiei − vi ∈ SpanC({tkej : k < min(µi, µj), 1 ≤ j ≤ m}) .

Hence item (ii) implies item (iii).


Finally, given item (iii), then we can see vi := tµiei −∑mj=1 pijej ∈ L for some pij ∈ C[t] of degree

less than min(µi, µj). It is easy to see that L = SpanO(v1, . . . , vm) and so L ∈ Wµ. To show that βµ

forms a basis for C[t]m/L, it suffices to show that for each i,

tµiei ∈ SpanC(tkei : 0 ≤ k < µi, 1 ≤ i ≤ m) + L .

This follows immediately from item (iii).

Given A ∈ Tµ, recall the definition of g(A) given in Equation (7.3.5). Note that g(A)t−µ ∈ G1(O∞).

Since g(A) ∈Mm(C[t])∩G1(O∞)tµ, we will regard g(A) as giving an element of Gr+∩Wµ. Alternatively,

we can see that g(A)C[t]m satisfies the condition of item (ii) from Lemma 7.4.3.

The following result is called the Mirkovic–Vybornov isomorphism [MV07b]. In its present form, it

can be found in [CK18, Theorem 3.2], except that we have tweaked both maps with a matrix transpose

[ ]tr.

Theorem 7.4.4. The map Tµ → Gr+ ∩Wµ given by A 7→ g(A)C[t]m is an isomorphism with inverse

given by

L 7→ [t|C[t]m/L]trβµ .

7.4.3 Upper-triangularity and the Mirkovic-Vybornov isomorphism

For the next result, we will consider the “intersection” of Grλ′,λ′′

0,A withWµ. As Grλ′,λ′′

0,A is not a subscheme

of Gr, by this intersection, we really mean the preimage of Wµ under the composition

Grλ′,λ′′

0,A ↪→ Gr0,A → Gr .

This is not a very serious abuse of notation, since the map Gr0,A → Gr is almost injective. In a similar

way, we will write Oλ′,λ′′

0,A ∩ Tµ using the “almost injective” map Oλ′,λ′′

0,A →MN (C).

The following refinement of the Mirkovic–Vybornov isomorphism is a special case of [MV07b, Theo-

rem 5.3].

Theorem 7.4.5. There is an isomorphism

Oλ′,λ′′

0,A ∩ Tµ ∼= Grλ′,λ′′

0,A ∩Wµ

given by (A, s) 7→ (g(A)C[t]m, s).

Proof. Since we already have the isomorphism from Theorem 7.4.4, it suffices to show that for any

A ∈ Tµ,

(A, s) ∈ Oλ′,λ′′

0,A ∩ Tµ if and only if (g(A)C[t]m, s) ∈ Grλ′,λ′′

0,A ∩Wµ .

This follows immediately from Lemma 7.4.2.

Theorem 7.4.6. The isomorphism from Theorem 7.4.5 restricts to an isomorphism

Oλ′,λ′′

0,A ∩ Uµ′,µ′′

0,A∼= Gr

λ′,λ′′

0,A ∩ Sµ′,µ′′

0,A .


Proof. We could prove this by observing the both sides are the attracting locus of an appropriate C×

action. However, we will give the following more algebraic proof.

Let A ∈ Tµ and s ∈ C. We must show that (A, s) ∈ Uµ′,µ′′

0,A if and only if (g(A)C[t]m, s) ∈ Sµ′,µ′′

0,A .

On the one hand, if (A, s) ∈ Uµ′,µ′′

0,A , then g(A) is lower-triangular with diagonal tµ′(t− s)µ′′ , and so

g(A) ∈ N−[t, t−1, (t− s)−1]tµ′(t− s)µ′′ .

On the other hand, if (g(A)C[t]m, s) ∈ Sµ′,µ′′

0,A , then we can write

gtµr = ntµ′(t− s)µ

′′

for some r ∈ G(C[t]), n ∈ N−(C[t, t−1, (t− s)−1]) and g = g(A)t−µ. Let h = (t− s)µ′′t−µ′′ which lies in

T1(O∞). Note that h−1nh ∈ N−(K∞), so we can factor it as h−1nh = n1n2, where n1 ∈ N−,1(O∞), n2 ∈N−(C[t]). So then after doing a bit of algebra, we reach

tµr(t−µn−12 tµ)t−µ = g−1hn1.

Since g, h, n1 ∈ G1(O∞), the right hand side g−1hn1 lies in G1(O∞). Since µ is dominant, t−µn2tµ ∈

N−(C[t]), and so the left hand side lies in tµG(C[t])t−µ.

Moreover, since µ is dominant, we know that

tµG(C[t])t−µ ∩G1(O∞) = N−,1(O∞) .

Thus, we deduce that g−1hn1 ∈ N−,1(O∞) and hence g(A) ∈ tµ′(t− s)µ′′N−,1(O∞). Since A ∈ Tµ, this

implies that (A, s) ∈ Uµ′,µ′′

0,A as desired.

Restricting to the zero fibre and applying θ0 together with Proposition 7.3.1, we recover the following

result from [Dra20a].

Corollary 7.4.7. ([Dra20a, Corollary 5.2.2]) The map A 7→ g(A)Om gives an isomorphism

Oλ ∩ Tµ ∩ n ∼= Grλ ∩ Sµ− .

7.5 Application to the Fusion Product

7.5.1 MV cycles and tableaux

Continuing now with the notation of the previous sections, we add the assumption that µ ≤ λ and forget

the assumption that µ is dominant. Let r = m(m− 1)/2 be the number of positive roots in GLm. We

will now explain how to index MV cycles for GLm using Young tableaux.

Denote by [a, b] the interval {a, a+ 1, . . . , b− 1, b} ⊂ N. We begin with the following definitions.

Definition 7.5.1. Let L ⊂ Om be a point in Gr+. We define the relative dimension of L by

rdimL := dimCOm/L .

If γ ⊂ [1,m], we define two lattices Lγ , Lγ in Oγ := SpanO(ei : i ∈ γ) by

Lγ := L ∩ Oγ Lγ := L/Lγc


where γc := [1,m] \ γ.

We will also need the analogous definitions when O is replaced by C[t] and L ∈ Gr+. We record the

following easy observations.

Lemma 7.5.1. Let L ∈ Gr+ or Gr+.

1. For any γ ⊂ [m], we have

rdimLγ + rdimLγc

= rdimL .

2. For any γ1 ⊂ γ2 ⊂ [m], we have

(Lγ2)γ1 = Lγ1 .

For any L ∈ Gr+ and γ ⊂ [m], we define Dγ(L) := rdimLγ = rdimL− rdimLγc . By [Kam10, Propo-

sition 9.3], this coincides with the definition of Dγ used to compute the hyperplanes of the associated

MV polytopes.

Lemma 7.5.2. Let L ∈ Gr+. The following are equivalent:

1. L ∈ Sµ+

2. rdimL[1,i] = µ1 + · · ·+ µi for all i = 1, . . . ,m.

3. rdimL[i+1,m] = µi+1 + · · ·+ µm for i = 1, . . . ,m.

Similarly, the following are equivalent:

1’. L ∈ Sµ−

2’. rdimL[i+1,m] = µi+1 + · · ·+ µm for all i = 1, . . . ,m.

3’. rdimL[1,i] = µ1 + · · ·+ µi for i = 1, . . . ,m.

Proof. We prove the first statement as the second is similar. Let L = ntµC[t]m for some n ∈ N+(K) and

let v1, . . . , vm denote the columns of the matrix ntµ. Then L = SpanO(v1, . . . , vm). Fix i ≤ m. Then

L[1,i] = SpanO(w1, . . . , wi) where wj denotes the first i entries of vj (by upper-triangularity, the rest of

the entries are 0, in any case).

Assume that L ⊂ Om. Then L[1,i] ⊂ Oi and rdimL[1,i] is the valuation of the determinant of

the matrix whose columns are w1, . . . , wi. Since this matrix is upper-triangular with diagonal entries

tµ1 , . . . , tµi , this determinant is tµ1+···+µi and hence rdimL[1,i] = µ1 + · · ·+ µi.

Thus item 1 implies item 2. The converse follows from the fact that every L lies in some Sν+ and by

the above reasoning, ν is determined by the values of dimL[1,i].

The equivalence of item 2 and item 3 follows from Lemma 7.5.1.

We now introduce some notation related to Young tableaux. Denote by Y T (λ) the set of Young

tableaux of shape λ and by Y T (λ)µ the subset of Y T (λ) of tableaux having weight µ. Let Y T =⋃λ Y T (λ) and denote by Y T+ ⊂ Y T the subset of those tableaux having dominant weight.

Given τ ∈ Y T (λ)µ and i ∈ {1, . . .m}, denote by λ(i) (resp. µ(i)) the shape (resp. the weight) of τ(i),

the tableau got from τ by discarding all boxes of weight exceeding i. Note that µ(i) only depends on µ,

while λ(i) depends on the tableau τ . We will regard λ(i) (resp. µ(i)) as an effective dominant coweight

(resp. effective coweight) for GLi.


The Lusztig datum n•(τ) of the tableau τ is a list of r non-negative integers defined from its Gelfand–

Tsetlin pattern (see [BZ88, Sect. 4]) gt(τ) = (λ(i)j)1≤j≤i≤m by the formula

n•(τ)(a,b) = λ(b)a − λ(b− 1)a = # of boxes on row a of τ of weight b

(a, b) = (1, 2), . . . , (1,m), (2, 3), . . . , (2,m), . . . , (m− 1,m) .

Note that λ−µ =∑

1≤a<b≤m

n•(τ)(a,b)βa,b where βa,b denotes the positive root of G with a 1 in the a slot

and a −1 in the b slot. The pattern gt(τ) is recorded as a lower-triangular matrix (the array of shapes

of subtableaux τ(i)) and the datum n•(τ) is recorded as a sequence, unless noted otherwise. Below is

an example with λ = (4, 2) and µ = (3, 2, 1).

τ = 1 1 1 22 3

gt(τ) =

3

4 1

4 2 0

n•(τ) = (1, 0, 1)

We can associate to τ the locus

Z(τ) = {L ∈ Sµ− : L[1,i] ∈ Grλ(i) for i = 1, . . .m} .

The following result is closely related to Theorem 5.4.3 from [Dra20a]. It is also closely related

to the description of MV cycles in terms of Kostant data obtained by Anderson–Kogan [AK04] (see

[Kam10, Section 9]). We remark that a different map from Young tableaux to MV cycles was obtained

by Gaussent, Littelmann and Nguyen [GLN13, Theorem 2]; we are not certain of the relation with our

construction.

Proposition 7.5.3. Z(τ) has a unique irreducible component of dimension ρ(λ− µ). Let Z(τ) denote

the closure of this component. Then, Z(τ) is the MV cycle whose Lusztig datum (with respect to the

standard reduced word) is n•(τ).

Remark 7.5.2. We believe that Z(τ) is irreducible, so that in fact Z(τ) = Z(τ).

Proof. The proof follows the same strategy as in [Dra20a]. First, we consider

Z(τ)1 := {L ∈ Sµ− ∩ Gr+ : L[1,i] ∈ Sλ(i)+ for i = 1, . . .m} .

Using Lemma 7.5.1(2) and Lemma 7.5.2, we see that

Z(τ)1 = {L ∈ Gr+ : rdimL[a,b] = λ(b)a + · · ·+ λ(b)b for all 1 ≤ a ≤ b ≤ m}

where note that λ(b)a + · · ·+ λ(b)b is the number of boxes on rows a, . . . , b of weight 1, . . . , b.

From the proof of [Kam10, Proposition 9.6], we see that Z(τ)1 is equal to tµA(n•(τ)), where A(n•)

is defined in [Kam10, Sect. 4.3]. In particular, it is irreducible and of dimension ρ(λ− µ). Moreover, its

closure is the MV cycle whose Lusztig datum is n•(τ). Now, by results of [Kam08] (see especially the

proof of Theorem 1.4), we note that Z(τ)1 ∩ Grλ is dense in Z(τ)1.

Fix i ∈ {1, . . . ,m}. If L ∈ Sµ−, then L[1,i] ∈ Sµ(i)− so we get a map

fi : Z(τ)1 → Sµ(i)− ∩ Sλ(i)

+ .


From the definition of Z(τ)1, we see that fi(Z(τ)1) = Z(τ(i))1. From above, fi(Z(τ(i))1) ∩ Grλ(i) is a

dense constructible subset of Z(τ(i))1. Since Z(τ)1 is irreducible, this implies that f−1i (S

µ(i)− ∩ Sλ(i)

+ ∩Grλ(i)) is a dense constructible subset of Z(τ)1.

Working with all i at once, we conclude that

Z(τ)2 :=

m⋂i=1

f−1i (S

µ(i)− ∩ Sλ(i)

+ ∩ Grλ(i))

is a dense constructible subset of Z(τ)1. Thus Z(τ)2 = Z(τ)1.

On the other hand, by definition

Z(τ)2 ⊂ Z(τ).

Since dim Z(τ)2 = ρ(λ − µ), we see that Z(τ) must have at least one component of the maximal

dimension. To see that it cannot have any other components of maximal dimension, we note that⊔τ∈Y T (λ)µ

Z(τ) ⊂ Grλ ∩ Sµ−

and the number of irreducible components of the right hand side equals |Y T (λ)µ|, thus on the left hand

side, each Z(τ) can only have one irreducible component of dimension ρ(λ− µ).

7.5.2 Fusion of MV cycles via fusion of generalized orbital varieties

Given A ∈MN (C) we denote by A∣∣Cp the restriction of A to the subspace spanned by the first p standard

basis vectors of CN . If A∣∣Cp(Cp) ⊂ Cp then we identify it with the p× p upper-left submatrix of A.

Let τ ∈ Y T (λ)µ with µ dominant. We define

X(τ) = {A ∈ Tµ ∩ n : A∣∣C|µ(i)| ∈ Oλ(i) for each i = 1, . . . ,m} .

Lemma 7.5.4. Under the Mirkovic–Vybornov isomorphism X(τ) is mapped isomorphically onto Z(τ).

Proof. Fix A ∈ X(τ) and i ∈ {1, . . . ,m}. Let Oi ⊂ Om denote the submodule generated by the

first i standard basis vectors. Let l = g(A∣∣C|µ(i)|)O

i and L = g(A)Om. By definition of the map

A 7→ g(A), since A is upper-triangular we have that l = L[1,i]. Moreover, since elements of Tµ ∩ n

are upper triangular A∣∣C|µ(i)| ∈ Tµ(i) ∩ ni where ni denotes the subalgebra of upper-triangular matrices

in M|µ(i)|(C). By definition of X(τ) this principal submatrix has Jordan type λ(i). It follows that

l = L[1,i] ∈ Grλ(i) ∩ Sµ(i) for each i = 1, . . . ,m so that L ∈ Z(τ).

Conversely, let L ∈ Z(τ). By Corollary 7.4.7, L = g(A)Om for some A ∈ Tµ ∩ n. Running the above

argument in reverse, we see that A ∈ X(τ).

By [Dra20a, Prop. 4.5.4] (or Proposition 7.5.3 above), X(τ) has a unique irreducible component of

dimension ρ(λ− µ). We write X(τ) for the closure of this component. It is an irreducible component of

Oλ ∩Tµ ∩ n and will be called a generalized orbital variety of type λ. The collection of all possible

generalized orbital varieties of type λ will be denoted X (λ).

Theorem 7.5.5. ([Dra20a, Theorem 4.8.2]) {X(τ) : τ ∈ Y T (λ)µ} is a complete set of irreducible

components of Oλ ∩ Tµ ∩ n.


Our next goal is to describe the fusion of two MV cycles using a “fusion” of generalized orbital

varieties, which we will now define. Let λ, λ′, λ′′ and µ, µ′, µ′′ be as in previous sections, once again

assuming that µ is dominant.

Given a point s ∈ A× and a pair of tableaux τ ′ ∈ Y T (λ′)µ′ and τ ′′ ∈ Y T (λ′′)µ′′ , we define

X(τ ′, τ ′′)0,s ={A ∈ Uµ

′,µ′′

0,s : A∣∣C|µ(i)| has Jordan type ((λ′(i), 0), (λ′′(i), s)) for i = 1, . . . ,m

}and

X(τ ′, τ ′′)0,A× ={

(A, s) ∈MN × A× : A ∈ X(τ ′, τ ′′)0,s

}.

Note that elements A ∈ X(τ ′, τ ′′) do not in general correspond to pairs (A′, A′′) ∈ X(τ ′)×X(τ ′′) because

if µ′ (resp. µ′′) is not dominant then X(τ ′) (resp. X(τ ′′)) is not well-defined.

Proposition 7.5.6. The image of X(τ ′, τ ′′)0,A× under the isomorphism of Theorem 7.4.6 is Z(τ ′) ∗A×Z(τ ′′).

Proof. Fix A ∈ X(τ ′, τ ′′)0,s and i ∈ {1, . . . ,m}. Let C[t]i ⊂ C[t]m = Cm ⊗C C[t] denote the C[t]-

submodule generated by the first i standard basis vectors.

The lattices l = g(A∣∣C|µ(i)|)C[t]i and L = g(A)C[t]m are related by the equation l = L[1,i] where recall

L[1,i] = (L+ C[t][i+1,m])/C[t][i+1,m] ⊂ C[t]m/C[t][i+1,m] = C[t]i.

By definition of Uµ′,µ′′

0,A we have A∣∣C|µ(i)| ∈ Uµ

′(i),µ′′(i)0,s and by definition of X(τ ′, τ ′′)0,s, the Jordan

type of A∣∣C|µ(i)| is ((λ′(i), 0), (λ′′(i), s)). So, l ∈ Grλ

′(i),λ′′(i)0,s by Theorem 7.4.6. On the other hand, since

L ∈ Sµ′,µ′′

0,s , l ∈ Sµ′(i),µ′′(i)

0,s .

Therefore, as i varies, we see that the pair (L(0), L(s)) ∈ Sµ′ × Sµ′′ is such that

(L[1,i](0), L[1,i](s)) = (L(0)[1,i], L(s)[1,i]) ∈ Grλ′(i) × Grλ

′′(i)

and so (L(0), L(s)) ∈ Z(τ ′)× Z(τ ′′).

Conversely, given L ∈ Z(τ ′)∗s Z(τ ′′), we can write L = g(A)C[t]m by Theorem 7.4.5. Then reversing

the above logic, we conclude that A ∈ X(τ ′, τ ′′)0,s.

Now, in analogy with the fusion of Section 7.2.2, we define X(τ ′, τ ′′)0,A to be the Zariski closure

of the unique top-dimensional component of X(τ ′, τ ′′)0,A× in MN × A. (As noted in remark 7.5.2, we

expect that all these varieties X(τ) ∼= Z(τ) are irreducible, which would mean that taking this “top-

dimensional component” irrelevant.) Then we define the fusion of generalized orbital varieties to be the

scheme-theoretic intersection

X(τ ′, τ ′′)0,0 = X(τ ′, τ ′′)0,A ∩MN × {0} in MN × A

By Proposition 7.3.1, it is contained in Oλ ∩ Tµ ∩ n.

We quickly recall the definition of intersection multiplicity that we will need following [Ful16, Exam-

ple 2.6.5]. We consider a fibre square of C-schemes

W V

D Y


with D an effective Cartier divisor and V an irreducible variety of dimension k. We assume that V is

not contained in the support of D. Let Z be an irreducible component of W ; it is a subvariety of V

of codimension 1. The multiplicity of Z in the intersection D ∩ V is defined to be the length of the

module OV,Z/(f) over the local ring OV,Z of V along Z, where f is a local equation of D|V on an affine

open subset of V which meets Z. Following [Ful16, chap. 7], this multiplicity is denoted by i(Z,D · V ).

For our purposes, the zero fibre in Gr0,A is an effective divisor D. For Z ′, Z ′′ subvarieties of Gr,V = Z ′ ∗A Z ′′ ⊂ Gr0,A is a variety not contained in the support of D. Then we choose an irreducible

component Z of W := D ∩ V = Z ′ ∗0 Z ′′. Thus, we may consider the intersection multiplicity i(Z,Z ′ ∗0Z ′′) := i(Z,D · V ).

In a similar way, we may consider the intersection multiplicities of generalized orbital varieties in

X(τ ′, τ ′′)0,0 = MN × {0} ∩X(τ ′, τ ′′)0,A.

Corollary 7.5.7. i(X(τ), X(τ ′, τ ′′)0,0) = i(Z(τ), Z(τ ′) ∗0 Z(τ ′′)) .

Proof. By the isomorphism of Lemma 7.5.4, X(τ) is a dense constructible subset of Z(τ). On the other

hand, Proposition 7.5.6 gives us an identification of X(τ ′, τ ′′)0,A× and Z(τ ′) ∗A× Z(τ ′′); the schemes

X(τ ′, τ ′′)0,A and Z(τ ′) ∗A Z(τ ′′) are constructed from these by taking the top-dimensional irreducible

component and then closure. Thus, we conclude that X(τ ′, τ ′′)0,A is a dense constructible subset of

Z(τ ′) ∗A Z(τ ′′). From the definition of intersection multiplicity, it is clear that the multiplicity can be

computed locally and so the result follows.

7.5.3 Multiplying MV basis elements in C[N ]

As GL∨m = GLm (so also N∨ = N) we will ignore the distinction between weights and coweights.

Recall that structure constants of multiplication in C[N ] with respect to the MV basis elements

bZ′ , bZ′′ ∈ Z(∞) are given by intersection multiplicities

bZ′ · bZ′′ =∑

Z∈Z(∞)

i (Z,Z ′ ∗0 Z ′′) bZ . (7.5.1)

Our next goal is to show that these structure constants can be computed using generalized orbital

varieties. Consider the following commutative diagram.

Y T+

⋃X (λ)

Y T⋃Z(λ)

⊕V (λ)

Nr Z(∞) C[N ]

τ 7→X(τ)

Lemma 7.5.4

τ 7→Z(τ)

n• t−λ Ψλσ

The sequence µ0 = (µ01, . . . , µ

0m−1) keeping track of the number of boxes of weight i in row i of a given

tableau is called its padding. Two tableaux have equal Lusztig data if and only if they are related by

padding, which is to say that we can change the padding of one (removing or adding boxes) to recreate

the other. Moreover, a tableau is completely determined by its Lusztig datum and padding.


For example

τ = 24

and τ =1 1 22 43

have equal Lusztig data, since we can increase the padding of τ (by adding to it the padding of τ) to

get τ , or forget the padding of τ to get τ .

We call a tableau stable if its padding cannot be decreased. The following lemma shows that stable

tableaux are in bijection with Lusztig data.

Lemma 7.5.8. Let n• = (n(a,b)) be a Lusztig datum and let µ0 = (µ0i ) be a padding. If λ = (

∑b n(a,b) +

µ0a)a then the smallest µ0 such that λ is dominant effective (and µ is effective) is:

µ0m, µ

0m−1 = 0

µ0i = max{0, µ0

i+1 +∑a

n(a,i+1) −∑a

n(a,i)} i = 1, . . . ,m− 2.

This choice of µ0 defines a section σ : Nr → Y T .

Proof sketch. To produce a tableau with Lusztig datum n• of smallest possible shape and weight:

• we can always take the number of m’s in row m, and the number of m − 1’s in row m − 1 to be

zero;

• we can take the number of m− 2’s in row m− 2 to be zero, unless n• tells us that there are more

boxes in row m− 1 than there are in row m− 2, and then we take µ0m−2 to offset the difference;

and iterate the last step up to µ01.

Corollary 7.5.9. Given two stable MV cycles Z ′, Z ′′ with Lusztig data n′•, n′′• resp. there exists a pair

of tableaux (τ ′, τ ′′) ∈ Y T (λ′)µ′ ×Y T (λ′′)µ′′ for some λ′, λ′′ dominant effective and some µ′, µ′′ effective,

such that µ = µ′ + µ′′ ≤ λ = λ′ + λ′′ are partitions and

bZ′ · bZ′′ =∑

τ∈Y T (λ)

i(X(τ), X(τ ′, τ ′′)0,0)bt−λZ(τ).

Proof. Choose τ ′ = σ(n′•), τ′′ = σ(n′′•) and suppose τ ′ ∈ Y T (λ′)µ′ , τ

′′ ∈ Y T (λ′′)µ′′ . Note that Z ′ =

t−λ′Z(τ ′), Z ′′ = t−λ

′′Z(τ ′′) so the intersection multiplicity of a stable MV cycle Z in Z ′ ∗0 Z ′′ can be

computed as the intersection multiplicity of an MV cycle Z(τ) in Z(τ ′) ∗0 Z(τ ′′) for some τ ∈ Y T (λ)µ,

because these two situations isomorphic by translation by tλ.

In case µ is not dominant, increase the padding of the larger of the two tableaux by (max(0, µ2 −µ1),max(0, µ3 − µ2), . . . ,max(0, µm − µm−1)).

Now, by Corollary 7.5.7, the intersection multiplicity of Z(τ) in Z(τ ′) ∗0 Z(τ ′′) can in turn be

computed as the intersection multiplicity of the generalized orbital variety X(τ) in X(τ ′, τ ′′)0,0. (We

have ensured µ is dominant, in order so that X(τ) will be defined.)

7.6 Examples

In this section, we will compute some examples of multiplication of MV basis elements using the method

provided by Corollary 7.5.9.


We use the section defined by Lemma 7.5.8 to abbreviate MV basis elements to tableaux and view

C[N ] as an algebra in these, rewriting Equation (7.5.1) as an equation in tableaux. The coefficients

are found using generalized orbital varieties. Suppose we have two tableaux τ ′ and τ ′′, with respective

weights λ′, µ′ and λ′′, µ′′, and we wish to form their (geometric) fusion product. Once we have applied

any necessary padding (as in the proof of Corollary 7.5.9) so that µ = µ′ + µ′′ is dominant, we take a

generic matrix A ∈ X(τ ′, τ ′′)0,s.

The requirement that Ai := A∣∣C|µ(i)| ∈ Oλ

′(i),λ′′(i)0,s for each i imposes rank conditions on A in the

form of vanishing and non-vanishing minors. We form the ideal I ⊂ C[Akij , s±] = C[Uµ

′,µ′′

0,A× ] generated by

these relations. The ideal of the fusion X(τ ′, τ ′′)0,0 is

J = (I ∩ C[Akij , s]) + (s) .

Taking the primary decomposition (to account for possible multiplicities) of J gives us ideals J1, . . . , Jn,

each corresponding to a generalized orbital variety occurring in the fusion. We use Theorem 7.5.5 to find

a tableau for each Ji and then forget all unnecessary padding to get the corresponding stable MV cycle.

Each Ji also tells us the multiplicity of each MV cycle in the fusion product, where we have multiplicity

1 if and only if Ji =√Ji.

Example 7.6.1. Consider the MV cycles in Example 1.2.1 and the exchange relation in Example 4.2.1.

The stable tableau for each MV cycle is given below:

Z1 = Z ( 2 ) Z2 = Z(

13

)Z+ = Z ( 3 ) Z− = Z

(23

)We will verify the exchange relation bZ1

bZ2= bZ+

+ bZ− through the fusion of Z1 and Z2. We have

the weights to be λ′ = (1, 0, 0), λ′′ = (1, 1, 0), µ′ = (0, 1, 0), and µ′′ = (1, 0, 1). The matrix under

consideration is therefore

A =

s A1

12 A113

0 A123

s

.There are no relations from the submatrices A1 and A2. From the submatrix A3 = A, we require

A112A

123 + sA1

13 = 0.

Then the ideal J is

(A112A

123, s) = (A1

12, s) ∩ (A123, s).

The corresponding tableaux are

Ideal of X(τ) τ

(A112, s)

1 32

(A123, s)

1 23

After removing the padding, we see that we get the tableaux for Z+ and Z−, and each has multiplicity


1 as the corresponding ideal is radical.

Example 7.6.2. 2 2 · 1 13 3

= 3 3 + 2 23 3

+ 2 · 2 33

: The requirement that

A =

0 1

−s2 2s A112 A2

12 A113 A2

13

0 1

0 A123 A2

23

0 1

−s2 2s

is contained in 2 2 ∗A× 1 1

3 3

imposes the following nontrivial relations on A.

2A212A

223s+ 3A2

13s2 +A2

12A123 +A1

12A223 + 2A1

13s,A213s

3 −A212A

123s−A1

12A223s− 2A1

12A123,

A212A

213A

123s

2 +A112A

213A

223s

2 − (A212A

123)

2 + 2A112A

212A

123A

223 − (A1

12A223)

2 + 6A112A

213A

123s+ 4A1

12A113A

123,

A112A

213(A

223)

2s2 − (A212A

123)

2A223 + 2A1

12A212A

123(A

223)

2 − (A112)

2(A223)

3 − 2A212A

213(A

123)

2s

+ 4A112A

213A

123A

223s−A1

13A213A

123s

2 − 3A112A

213(A

123)

2 + 4A112A

113A

123A

223,

(A212)

3(A123)

2A223 − 2A1

12A123(A

212A

223)

2 +A212(A

112)

2(A223)

3 + 2A213(A

212A

123)

2s+ 2A213(A

112A

223)

2s

+ 3A112(A

213)

2A123s

2 +A113(A

212A

123)

2 + 4A112A

212A

213(A

123)

2 − 6A112A

212A

113A

123A

223

+ 4(A112)

2A213A

123A

223 − 4A1

12A113A

213A

123s− 4A1

12(A113)

2A123 +A1

13(A112A

223)

2

At s = 0 the ideal generated by the above relations decomposes as a union of the following generalized

orbital varieties, where the last component occurs with multiplicity 2.

Ideal of X(τ) τ

(A112, A

212, s)

1 1 3 32 2

(A123, A

223, s)

1 1 2 23 3√

((A123)2, A2

12A123 +A1

12A223, A

112A

123, (A

112)2, s) 1 1 2 3

2 3

Chapter 8

A3 Case Study

In this chapter we will study the cluster structure in the A3 case.

8.1 Clusters in C[N ]

Our initial seed will be derived from the reduced expression i = (1, 2, 3, 1, 2, 1). We obtain the following

pieces of data:

k -3 -2 -1 1 2 3 4 5 6

ik -3 -2 -1 1 2 3 1 2 1

k+ 3 2 1 4 5 7 6 7 7

v>k w0 w0 w0 s1s2s1s3s2 s1s2s1s3 s1s2s1 s1s2 s1 e

with e(i) = {1, 2, 4}.The generalized minors are

∆(−3, i) =

∣∣∣∣∣∣∣x12 x13 x14

x22 x23 x24

x32 x33 x34

∣∣∣∣∣∣∣ ∆(1, i) = x13 ∆(4, i) = x12

∆(−2, i) =

∣∣∣∣∣x13 x14

x23 x24

∣∣∣∣∣ ∆(2, i) =

∣∣∣∣∣x12 x13

x22 x23

∣∣∣∣∣ ∆(5, i) =

∣∣∣∣∣x11 x12

x21 x22

∣∣∣∣∣∆(−1, i) = x14 ∆(3, i) =

∣∣∣∣∣∣∣x11 x12 x13

x21 x22 x23

x31 x32 x33

∣∣∣∣∣∣∣ ∆(6, i) = x11

with ∆(−3, i),∆(−2, i), and ∆(−1, i) as the frozen variables. The corresponding exchange quiver is

Ai =

−1 1 4 6

−2 2 5

−3 3

66

Chapter 8. A3 Case Study 67

so the exchange matrix is

Bi =

0 −1 0

−1 1 0

1 0 0

0 −1 1

1 0 −1

−1 1 0

0 1 0

0 −1 1

0 0 −1

where the columns are indexed by {1, 2, 4} and the rows are indexed by {−3,−2,−1, 1, 2, 4, 3, 5, 6}. As

the non-exchangeable elements are {3, 5, 6}, when we restrict to C[N ], the initial seed is

Σ =

∣∣∣∣∣∣∣x12 x13 x14

1 x23 x24

0 1 x34

∣∣∣∣∣∣∣ ,∣∣∣∣∣x13 x14

x23 x24

∣∣∣∣∣ , x14, x13,

∣∣∣∣∣x12 x13

1 x23

∣∣∣∣∣ , x12

,

0 −1 0

−1 1 0

1 0 0

0 −1 1

1 0 −1

−1 1 0

After successively mutating this seed, there are a total of 14 seeds and 12 total cluster variables. The

diagram below illustrates the cluster structure.

Each cluster has exactly 3 mutable variables, which constitute the vertices of a triangle and the initial

seed Σ corresponds to the outer triangle. For a seed corresponding to the triangle with vertices a, b, c,


mutation in say direction a will correspond to the unique other triangle that has b and c as its vertices.

The mutation of a is given by the third vertex of the triangle. For example, if we mutate Σ in direction

x12, then from the diagram we have µ(x12) = x23. Note that every cluster variable except x13x34 − x14

corresponds to a generalized minor.

Remark 8.1.1. Originally noticed by Anderson, a similar diagram to the above can be built by looking

at MV polytopes in type A3. We create a graph where each vertex represents a prime MV polytope,

that is, an MV polytope that can not be written as the sum of two MV polytopes, and an edge joins

MV polytopes P and Q if P + Q is also an MV polytope. The resulting graph will be the same as the

diagram above except it will have an additional edge, which joins the polytopes corresponding to the

polynomials x12x24− x14 and x13x34− x14. The reason for this is because the associated Λ-modules are

M1 = C C C1

0

0

1and M2 = C C C

0

1

1

0, and we have

Pol(M1) + Pol(M2) = Pol(S1) + Pol(S3) + Pol(P2).

The polynomials in C[N ] corresponding to the Λ-modules S1, S3, and P2 are, respectively, x12, x34, and

x13x24 − x14x23, which are all in one cluster. Hence even though M1 and M2 are not both in a cluster,

the sum of their polytopes is still an MV polytope as it can be written as a sum of polytopes of other

modules that are all in a cluster.

8.2 Clusters in Λ-mod

We will use the quiver Q = 1 2 3 . The universal quiver Λ is then

...

(1, 1) (1, 2) (1, 3)

(0, 1) (0, 2) (0, 3)

(−1, 1) (−1, 2) (−1, 3)

...

The subquiver ΓQ is then

(2, 1)

(1, 1) (1, 2)

(0, 1) (0, 2) (0, 3)

The ordering of the vertices of ΓQ is (0, 1) < (0, 2) < (0, 3) < (1, 1) < (1, 2) < (2, 1) so applying the

pushdown functor to this ordering, we get the reduced expression i = (1, 2, 3, 1, 2, 1) that is adapted to


Q. The injective indecomposables of CΓQ are

I(0,1) =

0

0 0

C C C1 1

I(0,2) =

0

C C

0 C C

1

1

1

1

I(0,3) =

C

0 1

0 0 C

1

1

I(1,1) =

0

C C

0 0 0

1 I(1,2) =

C

0 C

0 0 0

1

I(2,1) =

C

0 0

0 0 0

Hence the basic maximal rigid Λ-module is

T =

(C C C

0

1

0

1

)⊕

C C2 C

[0

1

][1 0]

[1 0][0

1

]⊕ ( C C C

1

0

1

0

)

⊕(

C C 00

1

0

0

)⊕(

C C 01

0

0

0

)⊕(

C 0 00

0

0

0

)Labelling the indecomposable summands of T = T1 ⊕ · · · ⊕ T6, we have the matrix

CT = (dim HomΛ(Ti, Tj)) =

1 1 1 0 0 0

1 2 1 1 1 0

1 1 1 1 1 1

1 1 0 1 1 0

1 1 0 1 1 1

1 0 0 1 0 1

Removing the three columns of −C−tT corresponding to the projective indecomposables, so the first three,

we obtain the matrix

B(T ) =

1 0 0

−1 1 0

0 −1 0

0 −1 1

1 0 −1

−1 1 0

After reordering the cluster variables, as ∆(−3, i) corresponds to FI(0,1) and ∆(−1, i) corresponds to

FI(0,3), we obtain the same matrix, and hence same seed, as the one computed for C[N ].

The 12 cluster modules along with their corresponding element in C[N ] are presented below:


Module in Λ-mod Corresponding element in C[N ]

S1 = C 0 00

0

0

0x12

S2 = 0 C 00

0

0

0x23

S3 = 0 0 C0

0

0

0x34

1← 2 := C C 00

1

0

0x13

1→ 2 := C C 01

0

0

0x12x23 − x13

2← 3 := 0 C C0

0

0

1x24

2→ 3 := 0 C C0

0

1

0x23x34 − x24

1→ 2← 3 := C C C1

0

0

1x12x24 − x14

1← 2→ 3 := C C C0

1

1

0x13x34 − x14

P1 = C C C1

0

1

0x12x23x34 − x12x24 − x13x34 + x14

P2 = C C2 C

[0

1

][1 0]

[1 0][0

1

] x13x24 − x14x23

P3 = C C C0

1

0

1x14

We mentioned that the element in C[N ] corresponding to the module 1 ← 2 → 3 was the only one

that is not a generalized minor. In addition to this, all the cluster modules are of the form N(γ) for

some γ ∈ Γ except 1← 2→ 3.

There are a total of 15 pairs of exchange sequences which we present only the three middle terms of

below:

S1 ↪→ (1← 2) � S2 S2 ↪→ (1→ 2) � S1

S2 ↪→ (2← 3) � S3 S3 ↪→ (2→ 3) � S2

(1← 2) ↪→ P3 ⊕ (1→ 2) � (1→ 2← 3) (1→ 2← 3) ↪→ S1 ⊕ P2 � (1← 2)

(1→ 2) ↪→ S1 ⊕ P2 � (1← 2→ 3) (1← 2→ 3) ↪→ P1 ⊕ (1← 2) � (1→ 2)

(1← 2) ↪→ S2 ⊕ P3 � (2← 3) (2← 3) ↪→ P2 � (1← 2)

(1→ 2) ↪→ P2 � (2→ 3) (2→ 3) ↪→ S2 ⊕ P1 � (1→ 2)

(1→ 2) ↪→ (1→ 2← 3) � S3 S3 ↪→ P1 � (1→ 2)

(1← 2) ↪→ P3 � S3 S3 ↪→ (1← 2→ 3) � (1← 2)

(2← 3) ↪→ (1→ 2← 3) � S1 S1 ↪→ P3 � (2← 3)

(2→ 3) ↪→ P1 � S1 S1 ↪→ (1← 2→ 3) � (2→ 3)

(1→ 2← 3) ↪→ P2 � S2 S2 ↪→ (1→ 2)⊕ (2← 3) � (1→ 2← 3)

(1← 2→ 3) ↪→ (1← 2)⊕ (2→ 3) � S2 S2 ↪→ P2 � (1← 2→ 3)

(2← 3) ↪→ S3 ⊕ P2 � (1← 2→ 3) (1← 2→ 3) ↪→ P3 ⊕ (2→ 3) � (2← 3)

(2→ 3) ↪→ P1 ⊕ (2← 3) � (1→ 2← 3) (1→ 2← 3) ↪→ S3 ⊕ P2 � (2→ 3)

(1→ 2← 3) ↪→ S1 ⊕ P2 ⊕ S3 � (1← 2→ 3) (1← 2→ 3) ↪→ P1 ⊕ P3 � (1→ 2← 3)


8.3 The MV Polytopes

We list each cluster module and its corresponding MV polytope where we label a point aα1 + bα2 + cα3

as (a, b, c):

Pol(S1) Pol(S2) Pol(S3)

(1, 0, 0)

0 (0, 1, 0)

0

(0, 0, 1)

0

Pol(1← 2) Pol(1→ 2) Pol(2← 3)

0

(1, 0, 0)

(1, 1, 0)

0

(0, 1, 0)

(1, 1, 0)

0(0, 1, 0)

(0, 1, 1)

Pol(2→ 3) Pol(1→ 2← 3) Pol(1← 2→ 3)

0

(0, 0, 1)

(0, 1, 1)

0

(0, 1, 0)

(0, 1, 1)

(1, 1, 0)

(1, 1, 1)

0(1, 0, 0)

(0, 0, 1)

(1, 0, 1)

(1, 1, 1)

Pol(P1) Pol(P2) Pol(P3)

0

(0, 0, 1)

(0, 1, 1)(1, 1, 1)

0

(0, 1, 0)

(0, 1, 1)

(1, 1, 0)

(1, 1, 1)

(1, 2, 1)

0(1, 0, 0)

(1, 1, 0)

(1, 1, 1)

We have the following visualization of the exchange relation for MV Polytopes:

Pol(S1) + Pol(S2) = 0 = Pol(1← 2) ∪ Pol(1→ 2)

Pol(S2) + Pol(S3) =

0

= Pol(2← 3) ∪ Pol(2→ 3)

Pol(1← 2)+

Pol(1→ 2← 3)=

0

= Pol(P3 ⊕ (1→ 2)) ∪ Pol(S1 ⊕ P2)


Pol(1→ 2)+

Pol(1← 2→ 3)=

0

= Pol(P1 ⊕ (1← 2)) ∪ Pol(S1 ⊕ P2)

Pol(1← 2) + Pol(2← 3) =

0

= Pol(P2) ∪ Pol(S2 ⊕ P3)

Pol(1→ 2) + Pol(2→ 3) =

0

= Pol(P2) ∪ Pol(S2 ⊕ P1)

Pol(1→ 2) + Pol(S3) =

0

= Pol(P1) ∪ Pol(1→ 2← 3)

Pol(1← 2) + Pol(S3) =

0

= Pol(P3) ∪ Pol(1← 2→ 3)

Pol(2← 3) + Pol(S1) =

0

= Pol(P3) ∪ Pol(1→ 2← 3)

Pol(2→ 3) + Pol(S1) =

0

= Pol(P1) ∪ Pol(1← 2→ 3)

Pol(1→ 2← 3) + Pol(S2) =

0

= Pol(P2) ∪ Pol((2← 3)⊕ (1→ 2))

Pol(1← 2→ 3) + Pol(S2) =

0

= Pol(P2) ∪ Pol((2→ 3)⊕ (1← 2))


Pol(2← 3) + Pol(1← 2→ 3) =

0

=Pol(S3 ⊕ P2)∪

Pol(P3 ⊕ (2→ 3))

Pol(2→ 3) + Pol(1→ 2← 3) =

0

Pol(S3 ⊕ P2)∪Pol(P1 ⊕ (2← 3))

Pol(1→ 2← 3)+

Pol(1← 2→ 3)=

0

=Pol(S1 ⊕ P2 ⊕ S3)∪

Pol(P1 ⊕ P3)


8.4 The MV Cycles

All except one of the MV cycles corresponding to cluster variables happen to be toric varieties whose

MV polytopes are the same as their toric polytopes. The exception is the MV cycle corresponding to

the module P2.

Below we list each cluster module along with its stable tableau, the coordinate ring of its generalized

orbital variety, and a variety that the MV cycle is isomorphic to:

Λ-Module τ Coordinate ring of X(τ) MV Cycle

S1 2 C[A112] P1

S213

C[A112,A

113,A

123]

(A112,A

113)

P1

S3

124

C[A112,A

212,A

113,A

114,A

123,A

124,A

134]

(A112,A

212,A

113,A

114,A

123,A

124)

P1

1← 2 3C[A1

12,A113,A

123]

(A112)

P2

1→ 2 23

C[A112,A

113,A

123]

(A123)

P2

2← 3 14

C[A112,A

113,A

114,A

123,A

124,A

134]

(A112,A

113,A

123,A

134)

P2

2→ 3134

C[A112,A

113,A

114,A

123,A

124,A

134]

(A112,A

113,A

114,A

134)

P2

1→ 2← 3 24

C[A112,A

212,A

113,A

114,A

123,A

124,A

134]

(A112,A

113,A

123,A

124)

S(2, 4)

1← 2→ 31 324

C[A112,A

113,A

114,A

123,A

124,A

134]

(A112,A

134,A

113A

124−A1

23A114)

S(2, 4)

P1

234

C[A112,A

113,A

114,A

123,A

124,A

134]

(A123,A

124,A

134)

P3

P234

C[A112,A

113,A

114,A

123,A

124,A

134]

(A112,A

134)

Gr(2, 4)

P3 4C[A1

12,A113,A

114,A

123,A

124,A

134]

(A112,A

113,A

123)

P3

Table 8.1. The variety S(2, 4) denotes the non-smooth Schubert divisor in Gr(2, 4) the Grassmannianof planes in C4

In what follows we check that the exchange relations in C[N ], written in terms of stable tableaux,

are corroborated by the fusion of the corresponding MV cycles.


Example 8.4.1. 2 · 13

= 3 + 23


A =

s A1

12 A113

0 A123

s

is contained in 2 ∗A× 13

imposes the relation A112A

123 + sA1

13 on A and at s = 0 the ideal generated by this relation decomposes

as a union of the following generalized orbital varieties.

Ideal of X(τ) τ

(A112, s)

1 32

(A123, s)

1 23

Example 8.4.2. 13·

124

= 14

+134


A =

0 1

s A112 A1

13 A114

s A123 A1

24

0 A134

s

is contained in 1

3∗A×

124

results in the following relations on submatrices:

Submatrix Relations

A2 A112

A3 A113

A4 A114, A

123A

134 + sA1

24


orbital varieties.

Ideal of X(τ) τ

(A112, A

113, A

123, A

114, s)

1 12 43

(A112, A

113, A

114, A

134, s)

1 12 34


Example 8.4.3. 2 ·134

=234

+1 324


A =

s A112 A1

13 A114

0 A123 A1

24

s A134

s

is contained in 2 ∗A×

134


Submatrix Relations

A3 A112A

123 + sA1

13

A4 A134, A

112A

124 + sA1

14, A123A

114 −A1

13A124


orbital varieties.

Ideal of X(τ) τ

(A123, A

124, A

134, s)

1 234

(A112, A

134, A

123A

114 −A1

13A124, s)

1 324

Example 8.4.4. 2 ·1 12 43

= 4 + 24


A =

0 1

−s2 2s A112 A2

12 A113 A1

14

0 1

s A123 A1

24

s A134

s

is contained in 2 ∗A×

1 12 43


Submatrix Relations

A2 A112 + sA2

12

A3 A113, A

123

A4 A212A

124 + sA1

14, A112A

124 − s2A1

14



orbital varieties.

Ideal of X(τ) τ

(A112, A

212, A

113, A

123, s)

1 1 42 23

(A112, A

113, A

123, A

124, s)

1 1 22 43

Example 8.4.5.1 124

· 23

= 14

+134


A =

0 1

0 A112 A

212 A1

13 A114

0 1

s A123 A1

24

s A134

0

is contained in

1 124

∗A× 23


Submatrix Relations

A2 A112

A3 A123

A4 A124, A

113A

134 − sA1

14


orbital varieties.

Ideal of X(τ) τ

(A112, A

113, A

123, A

124, s)

1 1 22 43

(A112, A

123, A

124, A

134, s)

1 1 22 34


Example 8.4.6.124· 3 = 4 +

1 324


A =

0 A112 A1

13 A114

0 A123 A1

24

s A134

0

is contained in

124∗A× 3


Submatrix Relations

A2 A112

A4 A123A

134 − sA1

24, A113A

134 − sA1

14, A113A

124 −A1

23A114


orbital varieties.

Ideal of X(τ) τ

(A112, A

113, A

123, s)

1 423

(A112, A

134, A

113A

124 −A1

23A114, s)

1 324

Example 8.4.7. 1 12 4

· 3 = 1 43

+ 34


A =

0 1

0 A112 A1

13 A114

0 A123 A1

24

s A134

0

is contained in 1 1

2 4∗A× 3


Submatrix Relations

A2 A112

A4 A113A

134 − sA1

14


orbital varieties.


Ideal of X(τ) τ

(A112, A

113, s)

1 1 42 3

(A112, A

134, s)

1 1 32 4

Example 8.4.8.1 12 34

· 23

= 34

+1 23 34


A =

0 1

0 A112 A

212 A1

13 A213 A1

14

0 1

s A123 A

223 A1

24

0 1

s A134

0

is contained in1 12 34

∗A× 23


Submatrix Relations

A2 A112

A3 A212A

123 − sA1

13, A123 + sA2

23, A212A

223 +A1

13

A4 A134, A

212A

124 − sA1

14, A114A

123 −A1

13A124


orbital varieties.

Ideal of X(τ) τ

(A112, A

212, A

113, A

123, A

134, s)

1 1 32 2 43

(A112, A

123, A

124, A

134, A

212A

223 +A1

13, s)1 1 22 3 34


Example 8.4.9.1 1 32 24

· 13

= 34

+1 334

= 34

+134· 3 : The requirement that

A =

0 1

0 1

s A112 A

212 A1

13 A213 A1

14

0 1

0 A123 A

223 A1

24

0 1

s A134

0

is contained in1 1 32 24

∗A× 13


Submatrix Relations

A2 A112, A

212

A3 A113 + sA2

13

A4 A134, A

114A

123 −A1

13A124


orbital varieties.

Ideal of X(τ) τ

(A112, A

212, A

113, A

123, A

134, s)

1 1 1 32 2 43

(A112, A

212, A

113, A

114, A

134, s)

1 1 1 32 2 34

Example 8.4.10. 13· 2

4= 3

4+ 1 2

3 4: The requirement that

A =

0 A112 A1

13 A114

s A123 A1

24

0 A134

s

is contained in 1

3∗A× 2

4



Submatrix Relations

A3 A112A

123 − sA1

13

A4 A123A

134 + sA1

24, A112A

124 +A1

13A134


orbital varieties.

Ideal of X(τ) τ

(A112, A

134, s)

1 32 4

(A123, A

112A

124 +A1

13A134, s)

1 23 4

Example 8.4.11.1 1 324

· 23

= 2 34

+2 334


A =

0 1

0 A112 A

212 A1

13 A213 A1

14

0 1

s A123 A

223 A1

24

0 1

s A134

0

is contained in1 1 324

∗A× 23


Submatrix Relations

A2 A112

A3 A123 + sA2

23

A4 A134, A

113A

124 −A1

23A114


orbital varieties.

Ideal of X(τ) τ

(A112, A

113, A

123, A

134, s)

1 1 2 32 43

(A112, A

123, A

124, A

134, s)

1 1 2 32 34


Example 8.4.12. 24· 1

3= 2 4

3+ 2 3

4: The requirement that

A =

s A112 A1

13 A114

0 A123 A1

24

s A134

0

is contained in 2

4∗A× 1

3


Submatrix Relations

A4 A123A

134 − sA1

24


orbital varieties.

Ideal of X(τ) τ

(A123, s)

1 2 43

(A134, s)

1 2 34

Example 8.4.13.1 1 32 23 4

· 14

=1 434

+1 32 44


A =

0 1

0 1

s A112 A

212 A1

13 A213 A1

14 A214

0 1

0 A123 A

223 A1

24 A224

0 1

0 A134 A

234

0 1

s

is contained in1 1 32 23 4

∗A× 14


Submatrix Relations

A2 A112, A

212

A3 A113, A

123

A4 A134, A

213A

234 +A1

14 + sA214, A

114A

223 −A2

13A124



orbital varieties.

Ideal of X(τ) τ

(A112, A

212, A

113, A

213, A

123, A

114, A

134, s)

1 1 1 42 2 33 4

(A112, A

212, A

113, A

123, A

134, A

223A

234 +A1

24, A213A

234 +A1

14, A213A

124 −A2

23A114, s)

1 1 1 32 2 43 4

Example 8.4.14.1 12 33 4

· 24

=1 32 44

+1 23 44


A =

0 1

0 A112 A

212 A1

13 A213 A1

14 A214

0 1

s A123 A

223 A1

24 A224

0 1

0 A134 A

234

0 1

s

is contained in1 12 33 4

∗A× 24


Submatrix Relations

A2 A112

A3 A113, A

123, A

212A

223 + sA2

13

A4 A134, A

223A

234 +A1

24 + sA224, A

212A

224 +A2

13A234 +A1

14, A212A

124 − sA1

14,

A114A

223 −A2

13A124, A

213A

124A

234 +A1

14A124 + sA1

14A224


orbital varieties.

Ideal of X(τ) τ

(A112, A

212, A

113, A

123, A

134, A

223A

234 +A1

24,

A213A

234 +A1

14, A213A

124 −A2

23A114, s)

1 1 32 2 43 4

(A112, A

113, A

123, A

223, A

124,

A134, A

212A

224 +A2

13A234 +A1

14, s)

1 1 22 3 43 4


Example 8.4.15.1 1 1 32 23 4

· 24

=2 434

+1 2 32 44


A =

0 1

0 1

0 A112 A

212 A

312 A1

13 A213 A1

14 A214

0 1

0 1

s A123 A

223 A1

24 A224

0 1

0 A134 A

234

0 1

s

is contained in1 1 1 32 23 4

∗A× 24


Submatrix Relations

A2 A112, A

212

A3 A113, A

123

A4 A134, A

223A

234 +A1

24 + sA224, A

114A

223 −A2

13A124, A

213A

124A

234 +A1

14A124 + sA1

14A224


orbital varieties.

Ideal of X(τ) τ

(A112, A

212, A

113, A

123, A

223, A

124, A

134, s)

1 1 1 2 42 2 33 4

(A112, A

212, A

113, A

123, A

134, A

223A

234 +A1

24, A213A

234 +A1

14, A213A

124 −A2

23A114, s)

1 1 1 2 32 2 43 4

In conclusion, we obtain the following result:

Proposition 8.4.1. In type A3, every cluster variable in C[N ] is in the MV basis.

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by Yuguang (Roger) Bai