Partition Identities and Quiver Representationsweigandt/QuiverPart.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois at

Partition Identities and Quiver Representations

Anna Weigandt

University of Illinois at Urbana-Champaign

[email protected]

January 23rd, 2017

Based on joint work with Richard Rimanyi and Alexander Yong

arXiv:1608.02030

Anna Weigandt Partition Identities and Quiver Representations

Overview

This project provides an elementary explanation for a quantumdilogarithm identity due to M. Reineke.

We use generating function techniques to establish a relatedidentity, which is a generalization of the Euler-Gauss identity.

This reduces to an equivalent form of Reineke’s identity in type A.


Representations of Quivers


Quivers

A quiver Q = (Q0,Q1) is a directed graph with

vertices: i ∈ Q0

edges: a : i → j ∈ Q1

See notes of Brion ([Bri08])


The Definition

A representation of Q is an assignment of a:

vector space Vi to each vertex i ∈ Q0 and

linear transformation fa : Vi → Vj to each arrow ia−→ j ∈ Q1

dim(V ) = (dimVi )i∈Q0 is the dimension vector of V .


The Definition

A representation of Q is an assignment of a:

vector space Vi to each vertex i ∈ Q0 and

linear transformation fa : Vi → Vj to each arrow ia−→ j ∈ Q1

dim(V ) = (dimVi )i∈Q0 is the dimension vector of V .


Morphisms of Representations

A morphism of quiver representations h : (V , f )→ (W , g) is acollection of linear transformations {hi : i ∈ Q0} so that for everyarrow i

a−→ j :

Two representations are isomorphic if each hi is invertible.


The Representation Space

Fix d ∈ NQ0 , The representation space is

RepQ(d) :=⊕

ia−→j∈Q1

Mat(d(i),d(j)).

LetGLQ(d) :=

∏i∈Q0

GL(d(i)).

GLQ(d) acts on RepQ(d) by base change at each vertex.

Orbits of this action are in bijection with isomorphism classes of ddimensional representations.


Example 1

What parameterizes isomorphism classes?

(g1, g2) · (A) = g1Ag−12

A choice of dimension vector and the rank of the map.


Example 1


(g1, g2) · (A) = g1Ag−12



Example 1


(g1, g2) · (A) = g1Ag−12



Example 1


(g1, g2) · (A) = g1Ag−12



Example 2


(g) · (A) = gAg−1

Jordan form (up to rearranging blocks)


Example 2


(g) · (A) = gAg−1



Example 2


(g) · (A) = gAg−1



Simple Representations

Assume Q has no oriented cycles. The simple representations αi

are formed by assigning C to vertex i and 0 to all other vertices.


Indecomposable Representations

From V and W we can build a new representation V ⊕W .

V is indecomposable if writing V ∼= V ′ ⊕ V ′′ implies V ′ or V ′′ istrivial.

Indecomposable representations are the building blocks for allrepresentations.


Indecomposable Representations

From V and W we can build a new representation V ⊕W .

V is indecomposable if writing V ∼= V ′ ⊕ V ′′ implies V ′ or V ′′ istrivial.

Indecomposable representations are the building blocks for allrepresentations.


Dynkin Quivers

A quiver is Dynkin if its underlying graph is of type ADE :


Gabriel’s Theorem

Theorem ([Gab75])

Dynkin quivers have finitely many isomorphism classes ofindecomposable representations.

For type A, indecomposables V[i ,j] are indexed by intervals.


Gabriel’s Theorem

Theorem ([Gab75])

Dynkin quivers have finitely many isomorphism classes ofindecomposable representations.

For type A, indecomposables V[i ,j] are indexed by intervals.


Reineke’s Identities


The Quantum Dilogarithm Series

E(z) =∞∑k=0

qk2/2zk

(1− q)(1− q2) . . . (1− qk)

This is a generalization of the Rogers dilogarithm


The Quantum Dilogarithm Series

E(z) =∞∑k=0

qk2/2zk

(1− q)(1− q2) . . . (1− qk)

This is a generalization of the Rogers dilogarithm


The Quantum Exponential

Define the quantum exponential:

expq(z) =∞∑k=0

zk

[k]q!,

where

[k]q! = 1(1 + q)(1 + q + q2) · · · (1 + q + . . .+ qk−1)

is the q-factorial.

The quantum dilogarithm series is:

E(z) = expq

(q1/2

1− qz

)


The Quantum Exponential

Define the quantum exponential:

expq(z) =∞∑k=0

zk

[k]q!,

where

[k]q! = 1(1 + q)(1 + q + q2) · · · (1 + q + . . .+ qk−1)

is the q-factorial.

The quantum dilogarithm series is:

E(z) = expq

(q1/2

1− qz

)


The Pentagon Identity

Theorem (Schutzenberger, Faddeev-Volkov, Faddeev-Kashaev)

If x and y are q-commuting variables, i.e. yx = qxy, then:

E(x)E(y) = E(y)E(q−1/2xy)E(x)

This generalizes the classical pentagon identity.

Reineke extended this idea to define a family of identitiescorresponding to Dynkin quivers.
















The Quantum Algebra of a Quiver

Define the Euler form

χ : NQ0 × NQ0 → Z

χ(d1, d2) =∑i∈Q0

d1(i)d2(i) −∑

ia−→j∈Q1

d1(i)d2(j)


The Quantum Algebra of a Quiver

AQ is an algebra over Q(q1/2) with

generators:{yd : d ∈ NQ0}

multiplication:

yd1yd2 = q12λ(d1,d2)yd1+d2

whereλ(d1, d2) = χ(d2, d1)− χ(d1, d2)


Reineke’s Identity

Given a representation V , we’ll write dV as a shorthand for ddim(V ).

For a Dynkin quiver, it is possible to fix a choice of ordering on the

simple representations: α1, . . . , αn

indecomposable representations: β1, . . . , βN

so that

E(yα1) · · ·E(yαn) = E(yβ1) · · ·E(yβN) (1)

(Original proof given by [Rei10], see [Kel11] for exposition and asketch of the proof.)


Reformulation

Looking at the coefficient of yd on each side, this is equivalent tothe following infinite family of identities ([Rim13]):

n∏i=1

1

(q)d(i)=∑η

qcodimC(η)N∏i=1

1

(q)mβi(η)

.

where the sum is over orbits η in RepQ(d) and mβ(η) is themultiplicity of β in V ∈ η.

Here, (q)k = (1− q) . . . (1− qk) is the q-shifted factorial.


Lacing Diagrams


Lacing Diagrams

A lacing diagram ([ADF85]) L is a graph so that:

the vertices are arranged in n columns labeled 1, 2, . . . , n

the edges between adjacent columns form a partial matching.


The Role of Lacing Diagrams in Representation Theory

When Q is a type A quiver, a lacing diagram can be interpreted asa sequence of partial permutation matrices which form arepresentation VL of Q.

1

2

3

4

[ 1 0 0 00 1 0 0

],

0 0 00 0 01 0 00 1 0

,

1 00 00 0

See [KMS06] for the equiorientated case and [BR04] forarbitrary orientations.


The Dimension Vector

dim(L) records the number of vertices in each column of L. Notethat dim(L) = dim(VL).

In this lacing diagram,

dim(L) = (2, 4, 3, 2)


The Dimension Vector

dim(L) records the number of vertices in each column of L. Notethat dim(L) = dim(VL).

In this lacing diagram,

dim(L) = (2, 4, 3, 2)


Equivalence Classes of Lacing Diagrams

Two lacing diagrams are equivalent if one can be obtained fromthe other by permuting vertices within a column.

{Equivalence Classes of Lacing Diagrams}

↕

{Isomorphism Classes of Representations of Q}


Equivalence Classes of Lacing Diagrams

Two lacing diagrams are equivalent if one can be obtained fromthe other by permuting vertices within a column.

{Equivalence Classes of Lacing Diagrams}

↕

{Isomorphism Classes of Representations of Q}


Strands

A strand is a connected component of L.

m[i ,j](L) = |{strands starting at column i and ending at column j}|

Example: m[1,2](L) = 2


Strands





Strands





Strands





Strands

Strands record the decomposition of VL into indecomposablerepresentations:

VL ∼= ⊕V⊕m[i,j](L)[i ,j]

Example: VL = V⊕2[1,2] ⊕ V[2,3] ⊕ V[2,4] ⊕ V[3,3] ⊕ V[4,4]


Statistics on Lacing Diagrams

tki (L) = |{strands starting at column i which use column k}|

Example: t32 (L) = 2











ski (L) = m[i ,k−1](L)

Example: s42 (L) = 1



ski (L) = m[i ,k−1](L)

Example: s42 (L) = 1


The Durfee Statistic

Let L(d) = {[L] : dim(L) = d}.

Fix a sequence of permutations w = (w (1), . . . ,w (n)), so thatw (i) ∈ Si and w (i)(i) = i .

Define the Durfee statistic:

rw(L) =∑

1≤i<j≤k≤n

skw (k)(i)

(L)tkw (k)(j)

(L).


The Durfee Statistic

Let L(d) = {[L] : dim(L) = d}.

Fix a sequence of permutations w = (w (1), . . . ,w (n)), so thatw (i) ∈ Si and w (i)(i) = i .

Define the Durfee statistic:

rw(L) =∑

1≤i<j≤k≤n

skw (k)(i)

(L)tkw (k)(j)

(L).


Theorem (Rimanyi, Weigandt, Yong, 2016)

Fix a dimension vector d = (d(1), . . . ,d(n)) and let w be asbefore. Then

n∏k=1

1

(q)d(k)=∑

η∈L(d)

qrw(η)n∏

k=1

1

(q)tkk (η)

k−1∏i=1

[tki (η) + ski (η)

ski (η)

]q

Note:[j+k

k

]qis the q-binomial coefficient.


Generating Series for Partitions


Generating Series

Let S be a set equipped with a weight function

wt : S → N

so that|{s ∈ S : wt(s) = k}| <∞

for each k ∈ N.

The generating series for S is

G (S , q) =∑s∈S

qwt(s).


Partitions

An integer partition is an ordered list of decreasing integers:

λ = λ1 ≥ λ2 ≥ . . . ≥ λℓ(λ) ≥ 0

We will typically represent a partition by its Young diagram:


The Generating Series for Rectangles of Width k

∅ . . .

1 + qk + q2k + q3k + q4k + q5k + . . . =1

1− qk



∅ . . .

1 + qk + q2k + q3k + q4k + q5k + . . . =1

1− qk



∅ . . .

1 + qk + q2k + q3k + q4k + q5k + . . .

=1

1− qk



∅ . . .

1 + qk + q2k + q3k + q4k + q5k + . . . =1

1− qk



1

(q)∞=

∞∏k=1

1

(1− qk)=

∞∏k=1

(1 + qk + q2k + q3k + . . .)

q16 = q1 · 1 · q3·3 · 1 · 1 · q6 · 1 · 1 · . . .



1

(q)∞=

∞∏k=1

1

(1− qk)=

∞∏k=1

(1 + qk + q2k + q3k + . . .)

q16 = q1 · 1 · q3·3 · 1 · 1 · q6 · 1 · 1 · . . .



1

(q)∞=

∞∏k=1

1

(1− qk)

=∞∏k=1

(1 + qk + q2k + q3k + . . .)

q16 = q1 · 1 · q3·3 · 1 · 1 · q6 · 1 · 1 · . . .



1

(q)∞=

∞∏k=1

1

(1− qk)=

∞∏k=1

(1 + qk + q2k + q3k + . . .)

q16 = q1 · 1 · q3·3 · 1 · 1 · q6 · 1 · 1 · . . .



1

(q)∞=

∞∏k=1

1

(1− qk)=

∞∏k=1

(1 + qk + q2k + q3k + . . .)

q16 = q1 · 1 · q3·3 · 1 · 1 · q6 · 1 · 1 · . . .


Notation

Let R(j , k) be the set consisting of a single rectangularpartition of size j × k .

G (R(j , k), q) =

qjk

Let P(j , k) be the set of partitions constrained to a j × kbox. (Here, we allow j , k =∞).

G (P(j , k), q) = ??


Notation


G (R(j , k), q) = qjk


G (P(j , k), q) = ??


Notation


G (R(j , k), q) = qjk


G (P(j , k), q) =

??


Notation


G (R(j , k), q) = qjk


G (P(j , k), q) = ??


P(∞, k): Partitions With at Most k Columns

Idea: Truncate the product

1

(q)k=

k∏i=1

1

(1− qi )


P(∞, k): Partitions With at Most k Columns

Idea: Truncate the product

1

(q)k=

k∏i=1

1

(1− qi )


P(k ,∞): Partitions With at Most k Rows

Idea: Bijection via conjugation

1

(q)k=

k∏i=1

1

(1− qi )


P(k ,∞): Partitions With at Most k Rows

Idea: Bijection via conjugation

1

(q)k=

k∏i=1

1

(1− qi )


q-Binomial Coefficients

Ordinary Binomial Coefficient:If x and y commute,

(x + y)n =∑i+j=n

(i + j

i

)x iy j .

q-Binomial Coefficient:If x and y are q commuting (yx = qxy),

(x + y)n =∑i+j=n

[i + j

i

]q

x iy j .


q-Binomial Coefficients

Ordinary Binomial Coefficient:If x and y commute,

(x + y)n =∑i+j=n

(i + j

i

)x iy j .

q-Binomial Coefficient:If x and y are q commuting (yx = qxy),

(x + y)n =∑i+j=n

[i + j

i

]q

x iy j .


Combinatorial Interpretation of the q-Binomial Coefficient

(x + y)11 = . . .+ yyxyxyyxxyy + . . .



(x + y)11 = . . .+ yyxyxyyxxyy + . . .

= . . .+ q15x4y7 + . . .

x

y y x

y x

y y x

yy

G (P(i , j), q) =[i + j

i

]q



(x + y)11 = . . .+ yyxyxyyxxyy + . . .

= . . .+ q15x4y7 + . . .

x

y y x

y x

y y x

yy

G (P(i , j), q) =[i + j

i

]q



(x + y)11 = . . .+ yyxyxyyxxyy + . . .

= . . .+ q15x4y7 + . . .

x

y y x

y x

y y x

yy

G (P(i , j), q) =[i + j

i

]q

=(q)i+j

(q)i (q)j



(x + y)11 = . . .+ yyxyxyyxxyy + . . .

= . . .+ q15x4y7 + . . .

x

y y x

y x

y y x

yy

G (P(i , j), q) =[i + j

i

]q

=(q)i+j

(q)i (q)j



(x + y)11 = . . .+ yyxyxyyxxyy + . . .

= . . .+ q15x4y7 + . . .

x

y y x

y x

y y x

yy

G (P(i , j), q) =[i + j

i

]q

=(q)i+j

(q)i (q)j


Durfee Squares and Rectangles


Durfee Squares

The Durfee square D(λ) is the largest j × j square partitioncontained in λ.


Durfee Squares

The Durfee square D(λ) is the largest j × j square partitioncontained in λ.


Durfee Squares

P(∞,∞) ←→∪j≥0

R(j , j)× P(j ,∞)× P(∞, j)

Euler-Gauss identity:

1

(q)∞=

∞∑j=0

qj2

(q)j(q)j(2)

See [And98] for details and related identities.


Durfee Squares

P(∞,∞) ←→∪j≥0

R(j , j)× P(j ,∞)× P(∞, j)

Euler-Gauss identity:

1

(q)∞=

∞∑j=0

qj2

(q)j(q)j(2)

See [And98] for details and related identities.


Durfee Rectangles

D(λ,−1) D(λ, 0) D(λ, 4)

The Durfee Rectangle D(λ, r) is the largest s × (s + r)rectangular partition contained in λ.

1

(q)∞=

∞∑s=0

qs(s+r)

(q)s(q)s+r. (3)

([GH68])


Durfee Rectangles

D(λ,−1) D(λ, 0) D(λ, 4)

The Durfee Rectangle D(λ, r) is the largest s × (s + r)rectangular partition contained in λ.

1

(q)∞=

∞∑s=0

qs(s+r)

(q)s(q)s+r. (3)

([GH68])


Proof of the Main Theorem




n∏k=1

1

(q)d(k)=∑

η∈L(d)

qrw(η)n∏

k=1

1

(q)tkk (η)

k−1∏i=1


ski (η)

]q

Idea: We will interpret each side as a generating series for tuples ofpartitions. Giving a weight preserving bijection between these twosets proves the identity.




n∏k=1

1

(q)d(k)=∑

η∈L(d)

qrw(η)n∏

k=1

1

(q)tkk (η)

k−1∏i=1


ski (η)

]q

Idea: We will interpret each side as a generating series for tuples ofpartitions. Giving a weight preserving bijection between these twosets proves the identity.


The Left Hand Side

S = P(∞,d(1))× . . .× P(∞,d(n))

G (S , q) =n∏

i=1

G (P(∞,d(i)), q) =n∏

i=1

1

(q)d(i)


The Right Hand Side

T =∪

η∈L(d)

T (η)

T (η) = R(η)× P(η)

G (T , q) =∑

η∈L(d)

G (T (η), q) =∑

η∈L(d)

G (R(η), q)G (P(η), q)


The Right Hand Side

T =∪

η∈L(d)

T (η)

T (η) = R(η)× P(η)

G (T , q) =∑

η∈L(d)

G (T (η), q) =∑

η∈L(d)

G (R(η), q)G (P(η), q)


Rectangles

R(η) = {µ = (µki ,j) : µ

ki ,j ∈ R(skw (k)(i)

(η), tkw (k)(j)

(η)), 1 ≤ i < j ≤ k ≤ n}

Recall the Durfee Statistic:

rw(η) =∑

1≤i<j≤k≤n

skw (k)(i)

(η)tkw (k)(j)

(η)

G (R(η), q) = qrw(η)


Rectangles

R(η) = {µ = (µki ,j) : µ

ki ,j ∈ R(skw (k)(i)

(η), tkw (k)(j)

(η)), 1 ≤ i < j ≤ k ≤ n}

Recall the Durfee Statistic:

rw(η) =∑

1≤i<j≤k≤n

skw (k)(i)

(η)tkw (k)(j)

(η)

G (R(η), q) = qrw(η)


Partitions

For convenience, write skk (η) =∞.

P(η) = {ν = (νki ) : νki ∈ P(skw (k)(i)

(η), tkw (k)(i)

(η)), 1 ≤ i ≤ k ≤ n}

G (P(η), q) =n∏

k=1

1

(q)tkk (η)

k−1∏i=1


ski (η)

]q


Partitions

For convenience, write skk (η) =∞.

P(η) = {ν = (νki ) : νki ∈ P(skw (k)(i)

(η), tkw (k)(i)

(η)), 1 ≤ i ≤ k ≤ n}

G (P(η), q) =n∏

k=1

1

(q)tkk (η)

k−1∏i=1


ski (η)

]q


The map S → T

Let w = (1, 12, 123, 1234) and d = (8, 9, 11, 8).

λ = (λ(1), λ(2), λ(3), λ(4)) ∈ S


λ(1)

s21

t11


λ(1)

t11

s21


λ(2)

t11

s21


λ(2)

t11

s21


λ(2)

t11

s21


λ(2)

t21

s21


λ(2)

t21t22

s21


λ(3)

t21t22

s31


λ(3)

t21t22

s31


λ(3)

t21t22

s31


λ(3)

t31

s31


λ(3)

t31t32t33

s31

s32


λ(4)

t31t32t33


λ(4)

t31t32t33


λ(4)

t31t32t33


λ(4)

t41

s41


λ(4)

t41t42t43t44

s41s42

s43


These Parameters are Well Defined

Lemma

There exists a unique η ∈ L(d) so that ski (η) = ski and tkj (η) = tkjfor all i , j , k.


A Recursion

For any η,ski (η) + tki (η) = tk−1

i (η) (4)

The parameters defined by Durfee rectangles satisfy the sameequations:

t11 t21t22

s21

t31t32t33

s31

s32

t41t42t43t44s41s42s43

λ 7→ (µ,ν) ∈ T (η) ⊆ T


A Recursion

For any η,ski (η) + tki (η) = tk−1

i (η) (4)

The parameters defined by Durfee rectangles satisfy the sameequations:

t11 t21t22

s21

t31t32t33

s31

s32

t41t42t43t44s41s42s43

λ 7→ (µ,ν) ∈ T (η) ⊆ T


λ 7→ (µ,ν)

∅


λ 7→ (µ,ν)

∅


What happens with a different choice of w?


The map S → T

Let w = (1, 12, 213, 3124) and d = (8, 9, 11, 8).

λ = (λ(1), λ(2), λ(3), λ(4)) ∈ S


λ(1) w (1) = 1

s21

t11


λ(1) w (1) = 1

t11

s21


λ(2) w (2) = 12

t11

s21


λ(2) w (2) = 12

t11

s21


λ(2) w (2) = 12

t11

s21


λ(2) w (2) = 12

t21

s21


λ(2) w (2) = 12

t21t22

s21


λ(3) w (3) = 213

t21 t22

s31


λ(3) w (3) = 213

t21 t22

s31


λ(3) w (3) = 213

t21 t22

s31


λ(3) w (3) = 213

t31

s31


λ(3) w (3) = 213

t32t31t33

s32

s31


λ(4) w (4) = 3124

t31t32 t33


λ(4) w (4) = 3124

t31t32 t33


λ(4) w (4) = 3124

t31t32 t33


λ(4) w (4) = 3124

t41

s41


λ(4) w (4) = 3124

t41t42 t43t44

s41s42

s43


The Associated Lacing Diagram

versus


The Associated Lacing Diagram

versus


Connection with Reineke’s Identity(in type A)


Simplifying the Identity

Lets think about the blue terms:

1

(q)t11

[s21 + t21

s21

]q

[s31 + t31

s31

]q

[s41 + t41

s41

]q

=

(1

(q)t11

)((q)s21+t21

(q)s21(q)t21

)((q)s31+t31

(q)s31(q)t31

)((q)s41+t41

(q)s41(q)t41

)

=1

(q)s21(q)s31

(q)s41(q)t41

=1

(q)m[1,1](q)m[1,2]

(q)m[1,3](q)m[1,4]




1

(q)t11

[s21 + t21

s21

]q

[s31 + t31

s31

]q

[s41 + t41

s41

]q

=

(1

(q)t11

)((q)s21+t21

(q)s21(q)t21

)((q)s31+t31

(q)s31(q)t31

)((q)s41+t41

(q)s41(q)t41

)

=1

(q)s21(q)s31

(q)s41(q)t41

=1

(q)m[1,1](q)m[1,2]

(q)m[1,3](q)m[1,4]




1

(q)t11

[s21 + t21

s21

]q

[s31 + t31

s31

]q

[s41 + t41

s41

]q

=

(1

(q)t11

)((q)s21+t21

(q)s21(q)t21

)((q)s31+t31

(q)s31(q)t31

)((q)s41+t41

(q)s41(q)t41

)

=1

(q)s21(q)s31

(q)s41(q)t41

=1

(q)m[1,1](q)m[1,2]

(q)m[1,3](q)m[1,4]




1

(q)t11

[s21 + t21

s21

]q

[s31 + t31

s31

]q

[s41 + t41

s41

]q

=

(1

(q)t11

)((q)s21+t21

(q)s21(q)t21

)((q)s31+t31

(q)s31(q)t31

)((q)s41+t41

(q)s41(q)t41

)

=1

(q)s21(q)s31

(q)s41(q)t41

=1

(q)m[1,1](q)m[1,2]

(q)m[1,3](q)m[1,4]




1

(q)t11

[s21 + t21

s21

]q

[s31 + t31

s31

]q

[s41 + t41

s41

]q

=

(1

(q)t11

)((q)s21+t21

(q)s21(q)t21

)((q)s31+t31

(q)s31(q)t31

)((q)s41+t41

(q)s41(q)t41

)

=1

(q)s21(q)s31

(q)s41(q)t41

=1

(q)m[1,1](q)m[1,2]

(q)m[1,3](q)m[1,4]



Doing these cancellations yields the identity:

Corollary (Rimanyi, Weigandt, Yong, 2016)

n∏i=1

1

(q)d(i)=∑

η∈L(d)

qrw(η)∏

1≤i≤j≤n

1

(q)m[i,j](η).

which looks very similar to:

n∏i=1

1

(q)d(i)=∑η

qcodimC(η)N∏i=1

1

(q)mβi(η)

.



Doing these cancellations yields the identity:

Corollary (Rimanyi, Weigandt, Yong, 2016)

n∏i=1

1

(q)d(i)=∑

η∈L(d)

qrw(η)∏

1≤i≤j≤n

1

(q)m[i,j](η).

which looks very similar to:

n∏i=1

1

(q)d(i)=∑η

qcodimC(η)N∏i=1

1

(q)mβi(η)

.


A Special Sequence of Permutations

We associate permutations w(i)Q ∈ Si to Q as follows:

Let w(1)Q = 1 and w

(2)Q = 12.

If ai−2 and ai−1 point in the same direction, append i to

w(i−1)Q

If ai−2 and ai−1 point in opposite directions, reverse w(i−1)Q

and then append i

wQ := (w(1)Q , . . . ,w

(n)Q )

Example:

1 2 3 4 5 6

a1 a2 a3 a4 a5

wQ = (1, 12, 123, 3214, 32145, 541236)


The Geometric Meaning of rw(η)

The Durfee statistic has the following geometric meaning:


codimC(η) = rwQ(η)

The above statement combined with the corollary implies Reineke’squantum dilogarithm identity in type A.


References I

S. Abeasis and A. Del Fra.

Degenerations for the representations of a quiver of type Am.

Journal of Algebra, 93(2):376–412, 1985.

G. E. Andrews.

The theory of partitions.

2. Cambridge university press, 1998.

A. S. Buch and R. Rimanyi.

A formula for non-equioriented quiver orbits of type a.

arXiv preprint math/0412073, 2004.

M. Brion.

Representations of quivers.

2008.

P. Gabriel.

Finite representation type is open.

In Representations of algebras, pages 132–155. Springer, 1975.


References II

B. Gordon and L. Houten.

Notes on plane partitions. ii.

Journal of Combinatorial Theory, 4(1):81–99, 1968.

B. Keller.

On cluster theory and quantum dilogarithm identities.

In Representations of algebras and related topics, EMS Ser. Congr. Rep.,pages 85–116. Eur. Math. Soc., Zurich, 2011.

A. Knutson, E. Miller and M. Shimozono.

Four positive formulae for type A quiver polynomials.

Inventiones mathematicae, 166(2):229–325, 2006.

M. Reineke.

Poisson automorphisms and quiver moduli.

Journal of the Institute of Mathematics of Jussieu, 9(03):653–667, 2010.


References III

R. Rimanyi.

On the cohomological Hall algebra of Dynkin quivers.

arXiv:1303.3399, 2013.


Merci!



Documents

Partition Identities and Quiver Representationsweigandt/QuiverPart.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois at