icerm.brown.edu · Background Crystals Crystals (by example) 2 dim There are 6 one dimensional weight spaces and 1 two dimensional weight space. The generators F 1 and F 2 act between

Affine Mirkovic-Vilonen polytopes 1

Peter Tingley (Loyola-Chicago)

Includes work with T. Dunlap, P. Baumann, J. Kamnitzer, D. Muthiah, B. Webster

ICERM, March 5, 2013

1Slides available at http://webpages.math.luc.edu/∼ptingley/Peter Tingley (Loyola-Chicago) MV polytopes ICERM, March 5, 2013 1 / 26

Outline

1 BackgroundCrystalsAn application: tensor product rules

2 Three constructions of MV polytopesFrom PBW basesFrom quiver varietiesFrom categorification (KLR algebras)Sketch of a proof

3 Affine MV polytopesDefinitionsl2 polytopesProving they exist

Peter Tingley (Loyola-Chicago) MV polytopes ICERM, March 5, 2013 2 / 26

Background Crystals

Crystals (by example)

2 dim

There are 6 one dimensional weight spaces and 1 two dimensionalweight space.The generators F1 and F2 act between weight spaces.There are 4 distinguished one dimensional spaces in the middle.If we use Uq(sl3) and ‘rescale’ the operators, then ‘at q = 0’, they matchup. You get a colored directed graph.


Background Crystals


2 dim

Consider the adjoint representation of sl3 (i.e. sl3 acting on itself).

There are 6 one dimensional weight spaces and 1 two dimensionalweight space.The generators F1 and F2 act between weight spaces.There are 4 distinguished one dimensional spaces in the middle.If we use Uq(sl3) and ‘rescale’ the operators, then ‘at q = 0’, they matchup. You get a colored directed graph.


Background Crystals


2 dim

Consider the adjoint representation of sl3 (i.e. sl3 acting on itself).There are 6 one dimensional weight spaces and 1 two dimensionalweight space.

The generators F1 and F2 act between weight spaces.There are 4 distinguished one dimensional spaces in the middle.If we use Uq(sl3) and ‘rescale’ the operators, then ‘at q = 0’, they matchup. You get a colored directed graph.


Background Crystals


2 dim

Consider the adjoint representation of sl3 (i.e. sl3 acting on itself).There are 6 one dimensional weight spaces and 1 two dimensionalweight space.



Background Crystals


2 dim

Consider the adjoint representation of sl3 (i.e. sl3 acting on itself).There are 6 one dimensional weight spaces and 1 two dimensionalweight space.The generators F1 and F2 act between weight spaces.

There are 4 distinguished one dimensional spaces in the middle.If we use Uq(sl3) and ‘rescale’ the operators, then ‘at q = 0’, they matchup. You get a colored directed graph.


Background Crystals


1

1

1

1

2

2

2

2

2 dim

Consider the adjoint representation of sl3 (i.e. sl3 acting on itself).There are 6 one dimensional weight spaces and 1 two dimensionalweight space.The generators F1 and F2 act between weight spaces.



Background Crystals


1

1

1

1

2

2

2

2

2 dim

Consider the adjoint representation of sl3 (i.e. sl3 acting on itself).There are 6 one dimensional weight spaces and 1 two dimensionalweight space.The generators F1 and F2 act between weight spaces.There are 4 distinguished one dimensional spaces in the middle.

If we use Uq(sl3) and ‘rescale’ the operators, then ‘at q = 0’, they matchup. You get a colored directed graph.


Background Crystals


1

1

1

1

2

2

2

2

2 dim

Consider the adjoint representation of sl3 (i.e. sl3 acting on itself).There are 6 one dimensional weight spaces and 1 two dimensionalweight space.The generators F1 and F2 act between weight spaces.There are 4 distinguished one dimensional spaces in the middle.If we use Uq(sl3) and ‘rescale’ the operators, then ‘at q = 0’, they matchup.

You get a colored directed graph.


Background Crystals


1

1

1

1

2

2

2

2

Consider the adjoint representation of sl3 (i.e. sl3 acting on itself).There are 6 one dimensional weight spaces and 1 two dimensionalweight space.The generators F1 and F2 act between weight spaces.There are 4 distinguished one dimensional spaces in the middle.If we use Uq(sl3) and ‘rescale’ the operators, then ‘at q = 0’, they matchup.

You get a colored directed graph.


Background Crystals


1

1

1

1

2

2

2

2

Consider the adjoint representation of sl3 (i.e. sl3 acting on itself).There are 6 one dimensional weight spaces and 1 two dimensionalweight space.The generators F1 and F2 act between weight spaces.There are 4 distinguished one dimensional spaces in the middle.If we use Uq(sl3) and ‘rescale’ the operators, then ‘at q = 0’, they matchup. You get a colored directed graph.


Background Crystals


1

1

1

1

2

2

2

2

Often the vertices of the crystal graph can be parametrized bycombinatorial objects.



Background Crystals


1 12

1 22

1 1

1 23

1 32

2 23

1 33

2 33

1

1

1

1

2

2

2

2

3

Often the vertices of the crystal graph can be parametrized bycombinatorial objects.



Background Crystals


1 12

1 22

1 1

1 23

1 32

2 23

1 33

2 33

1

1

1

1

2

2

2

2

3

Often the vertices of the crystal graph can be parametrized bycombinatorial objects.Then the combinatorics gives information about representation theory,and vise-versa.



Background Crystals


1 12

1 22

1 1

1 23

1 32

2 23

1 33

2 33

1

1

1

1

2

2

2

2

3

Often the vertices of the crystal graph can be parametrized bycombinatorial objects.Then the combinatorics gives information about representation theory,and vise-versa.Here you see that the graded dimension of the representation is thegenerating function for semi-standard Young tableaux.

If we use Uq(sl3) and ‘rescale’ the operators, then ‘at q = 0’, they matchup. You get a colored directed graph.


Background An application: tensor product rules

An application: tensor product rules

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There is a simple tensor product rule for crystals, which correctlypredicts tensor product multiplicities for representations.For sl2, crystals are just directed segments.Lets take the tensor product of two of these.For other types, just treat each sl2 independently!As an example, consider V ⊗ V for sl3.




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There is a simple tensor product rule for crystals, which correctlypredicts tensor product multiplicities for representations.

For sl2, crystals are just directed segments.Lets take the tensor product of two of these.For other types, just treat each sl2 independently!As an example, consider V ⊗ V for sl3.




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∧2(V)

There is a simple tensor product rule for crystals, which correctlypredicts tensor product multiplicities for representations.For sl2, crystals are just directed segments.

Lets take the tensor product of two of these.For other types, just treat each sl2 independently!As an example, consider V ⊗ V for sl3.




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∧2(V)

There is a simple tensor product rule for crystals, which correctlypredicts tensor product multiplicities for representations.For sl2, crystals are just directed segments.

Lets take the tensor product of two of these.For other types, just treat each sl2 independently!As an example, consider V ⊗ V for sl3.




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There is a simple tensor product rule for crystals, which correctlypredicts tensor product multiplicities for representations.For sl2, crystals are just directed segments.Lets take the tensor product of two of these.

For other types, just treat each sl2 independently!As an example, consider V ⊗ V for sl3.




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There is a simple tensor product rule for crystals, which correctlypredicts tensor product multiplicities for representations.For sl2, crystals are just directed segments.Lets take the tensor product of two of these.For other types, just treat each sl2 independently!

As an example, consider V ⊗ V for sl3.




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The infinity crystal

Bω1+2ω2⋃Bω1+ω2

There is a crystal B(λ) for each dominant integral weight λ.

{B(λ)} forms a directed system.

The limit of this system is B(∞).


















Bω1+2ω2

⋃Bω1+ω2




































I am interested in realizing all this combinatorics explicitly.

That means understanding B(∞), and all the embeddingsB(λ) ↪→ B(∞), how all V(λ)⊗ V(µ) decompose...





I am interested in realizing all this combinatorics explicitly.

That means understanding B(∞), and all the embeddingsB(λ) ↪→ B(∞), how all V(λ)⊗ V(µ) decompose...


Three constructions of MV polytopes From PBW bases

PBW bases and parameterizations of B(−∞)

Each reduced expression w0 = si1si2 · · · siN , gives an order

αi1 = β1 < β2 < . . . < βN

on positive roots.

Lusztig defines corresponding elements Fβj in U−q (g)βj .

{F(n1)β1· · ·F(nN)

βN} is a crystal basis for U−(g) (the PBW basis).

In particular, these monomials index B(∞).

There is one parameterization of B(∞) for each expression for w0...it isnatural to ask how they are related.

In a sense MV polytopes give an answer: We record each monomial as apath in weight space, and this is the 1-skeleton of the MV polytope.





αi1 = β1 < β2 < . . . < βN

on positive roots.


{F(n1)β1· · ·F(nN)









αi1 = β1 < β2 < . . . < βN

on positive roots.


{F(n1)β1· · ·F(nN)









αi1 = β1 < β2 < . . . < βN

on positive roots.


{F(n1)β1· · ·F(nN)









αi1 = β1 < β2 < . . . < βN

on positive roots.


{F(n1)β1· · ·F(nN)









αi1 = β1 < β2 < . . . < βN

on positive roots.


{F(n1)β1· · ·F(nN)









αi1 = β1 < β2 < . . . < βN

on positive roots.


{F(n1)β1· · ·F(nN)







Example for sl3

−α1 −α2

In this case there are exactly two reduced expressions for w0:

i1 := s1s2s1 and i2 := s2s1s2.

One finds that, e.g.,

(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Example for sl3

−α1 −α2


i1 := s1s2s1 and i2 := s2s1s2.


(Fi1α2

)(1)(Fi1α1+α2

)(2)(Fi1α1

)(3) = (Fi2α1

)(4)(Fi2α1+α2

)(1)(Fi2α2

)(2) mod q.



Characterization of MV polytopes

−α1 −α2

µ2

µ1

µ2

µ1

It is natural to ask which polytopes show up in this way.For rank two cases, this can be done using tropical Plücker relations.Equivalently, the conditions can be given in terms of the two diagonals:• Both have slope at most the corresponding simple root.• For one of the diagonals, the slope is equal to the simple root.Remarkably, understanding rank 2 is enough! A polytope is MV exactlyif all its rank 2 faces are MV polytopes of the right types.




−α1 −α2

µ2

µ1

µ2

µ1





−α1 −α2

µ2

µ1

µ2

µ1

It is natural to ask which polytopes show up in this way.

For rank two cases, this can be done using tropical Plücker relations.Equivalently, the conditions can be given in terms of the two diagonals:• Both have slope at most the corresponding simple root.• For one of the diagonals, the slope is equal to the simple root.Remarkably, understanding rank 2 is enough! A polytope is MV exactlyif all its rank 2 faces are MV polytopes of the right types.




−α1 −α2

µ2

µ1

µ2

µ1

It is natural to ask which polytopes show up in this way.For rank two cases, this can be done using tropical Plücker relations.

Equivalently, the conditions can be given in terms of the two diagonals:• Both have slope at most the corresponding simple root.• For one of the diagonals, the slope is equal to the simple root.Remarkably, understanding rank 2 is enough! A polytope is MV exactlyif all its rank 2 faces are MV polytopes of the right types.




−α1 −α2

µ2

µ1

µ2

µ1

It is natural to ask which polytopes show up in this way.For rank two cases, this can be done using tropical Plücker relations.Equivalently, the conditions can be given in terms of the two diagonals:• Both have slope at most the corresponding simple root.• For one of the diagonals, the slope is equal to the simple root.

Remarkably, understanding rank 2 is enough! A polytope is MV exactlyif all its rank 2 faces are MV polytopes of the right types.




−α1 −α2

µ2

µ1

µ2

µ1






−α1 −α2

µ2

µ1

µ2

µ1






−α1 −α2

µ2

µ1

µ2

µ1






−α1 −α2

µ2

µ1

µ2

µ1




Properties of MV polytopes

−α1 −α2

All edges are parallel to roots (here α1, α2, α1 + α2).Any path from bottom to top that passes through exactly one edgeparallel to each root uniquely determines an MV polytope.The crystal operators fi just increase the length of the bottom edge in awell chosen path, and adjust the rest of the polytope accordingly.B(λ) ⊂ B(∞) is the set of polytopes contained in a fixed ambientpolytope.Tensor product multiplicities are given by counting MV polytopessubject to conditions on top and bottom edge lengths.

We’ll now look at several places these polytopes arise (quiver varietiesand KLR algebras, as well as PBW bases). These constructions makesense in affine type, and we use them to figure out what affine MVpolytopes should be.

.Peter Tingley (Loyola-Chicago) MV polytopes ICERM, March 5, 2013 9 / 26



−α1 −α2






−α1 −α2

All edges are parallel to roots (here α1, α2, α1 + α2).

Any path from bottom to top that passes through exactly one edgeparallel to each root uniquely determines an MV polytope.The crystal operators fi just increase the length of the bottom edge in awell chosen path, and adjust the rest of the polytope accordingly.B(λ) ⊂ B(∞) is the set of polytopes contained in a fixed ambientpolytope.Tensor product multiplicities are given by counting MV polytopessubject to conditions on top and bottom edge lengths.





−α1 −α2

All edges are parallel to roots (here α1, α2, α1 + α2).Any path from bottom to top that passes through exactly one edgeparallel to each root uniquely determines an MV polytope.

The crystal operators fi just increase the length of the bottom edge in awell chosen path, and adjust the rest of the polytope accordingly.B(λ) ⊂ B(∞) is the set of polytopes contained in a fixed ambientpolytope.Tensor product multiplicities are given by counting MV polytopessubject to conditions on top and bottom edge lengths.





−α1 −α2

All edges are parallel to roots (here α1, α2, α1 + α2).Any path from bottom to top that passes through exactly one edgeparallel to each root uniquely determines an MV polytope.The crystal operators fi just increase the length of the bottom edge in awell chosen path, and adjust the rest of the polytope accordingly.

B(λ) ⊂ B(∞) is the set of polytopes contained in a fixed ambientpolytope.Tensor product multiplicities are given by counting MV polytopessubject to conditions on top and bottom edge lengths.





−α1 −α2

All edges are parallel to roots (here α1, α2, α1 + α2).Any path from bottom to top that passes through exactly one edgeparallel to each root uniquely determines an MV polytope.The crystal operators fi just increase the length of the bottom edge in awell chosen path, and adjust the rest of the polytope accordingly.B(λ) ⊂ B(∞) is the set of polytopes contained in a fixed ambientpolytope.

Tensor product multiplicities are given by counting MV polytopessubject to conditions on top and bottom edge lengths.





−α1 −α2






−α1 −α2






−α1 −α2




Three constructions of MV polytopes From quiver varieties

Quiver varieties (for sl5)

Q =- - -

� � �i i i i1 2 3 4 .

The preprojective algebra is the path algebra of Q modulo the relations

∗i1 -� i2 = 0,

∗i1 -� i2 =

∗i2 -� i3 ,

∗i2 -� i3 =

∗i3 -� i4 ,

∗i3 -� i4 = 0.

Fix an I-graded vector space V of dimension v. Lusztig’s quiver varietyΛ(v) is the variety of representations of Λ on V .

Kashiwara-Saito showed that∐

v IrrΛ(v), the union of all irreduciblecomponents of all Λ(v), naturally realizes B(∞).

We often identify v with the element∑

I viαi in the root lattice.




Q =- - -

� � �i i i i1 2 3 4 .


∗i1 -� i2 = 0,

∗i1 -� i2 =

∗i2 -� i3 ,

∗i2 -� i3 =

∗i3 -� i4 ,

∗i3 -� i4 = 0.









Q =- - -

� � �i i i i1 2 3 4 .


∗i1 -� i2 = 0,

∗i1 -� i2 =

∗i2 -� i3 ,

∗i2 -� i3 =

∗i3 -� i4 ,

∗i3 -� i4 = 0.









Q =- - -

� � �i i i i1 2 3 4 .

The preprojective algebra is the path algebra of Q modulo the relations∗i1 -� i2 = 0,

∗i1 -� i2 =

∗i2 -� i3 ,

∗i2 -� i3 =

∗i3 -� i4 ,

∗i3 -� i4 = 0.









Q =- - -

� � �i i i i1 2 3 4 .


∗i1 -� i2 =

∗i2 -� i3 ,

∗i2 -� i3 =

∗i3 -� i4 ,

∗i3 -� i4 = 0.









Q =- - -

� � �i i i i1 2 3 4 .


∗i1 -� i2 =

∗i2 -� i3 ,

∗i2 -� i3 =

∗i3 -� i4 ,

∗i3 -� i4 = 0.









Q =- - -

� � �i i i i1 2 3 4 .


∗i1 -� i2 =

∗i2 -� i3 ,

∗i2 -� i3 =

∗i3 -� i4 ,

∗i3 -� i4 = 0.









Q =- - -

� � �i i i i1 2 3 4 .


∗i1 -� i2 =

∗i2 -� i3 ,

∗i2 -� i3 =

∗i3 -� i4 ,

∗i3 -� i4 = 0.









Q =- - -

� � �i i i i1 2 3 4 .


∗i1 -� i2 =

∗i2 -� i3 ,

∗i2 -� i3 =

∗i3 -� i4 ,

∗i3 -� i4 = 0.









Q =- - -

� � �i i i i1 2 3 4 .


∗i1 -� i2 =

∗i2 -� i3 ,

∗i2 -� i3 =

∗i3 -� i4 ,

∗i3 -� i4 = 0.








MV polytopes from quiver varieties

Theorem (basically due to Baumann-Kamnitzer)

Fix b ∈ B(∞) and let Zb be the corresponding component in some Λ(v).Fix π ∈ Zb generic, and let T = (π,V) be the corresponding representation.Then the MV polytope MVb is the convex hull of the dimension vectors of allsubrepresentations of T.










Fix b ∈ B(∞) and let Zb be the corresponding component in some Λ(v).

Fix π ∈ Zb generic, and let T = (π,V) be the corresponding representation.Then the MV polytope MVb is the convex hull of the dimension vectors of allsubrepresentations of T.





Fix b ∈ B(∞) and let Zb be the corresponding component in some Λ(v).Fix π ∈ Zb generic, and let T = (π,V) be the corresponding representation.

Then the MV polytope MVb is the convex hull of the dimension vectors of allsubrepresentations of T.







Three constructions of MV polytopes From categorification (KLR algebras)

KLR algebras and crystals

DefinitionThe type g Khovanov-Lauda-Rouquier algebras are a family of algebras R(ν)for ν =

∑i niαi a positive sum of simple roots.

There are inclusions R(ν)× R(γ) ↪→ R(ν + γ).

⊕νgrepR(ν), the direct sum of the cagteories of Z-gradedrepresentations, categorifies U−q (g).

The simple modules (up to grading shift) index the crystal B(∞) (andactually better, they are a canonical basis).











































Some generators and relations that probably won’t help

R(ν) can be described in terms of pictures of strings, with ni strings of eachcolor. In this way, the generators are things like

i1 i2 in

· · ·

ei

i1 ij in

· · ·· · ·

yj

i1 ij ij+1 in

· · ·· · ·

ψj

And there are some relations:




R(ν) can be described in terms of pictures of strings, with ni strings of eachcolor.

In this way, the generators are things like

i1 i2 in

· · ·

ei

i1 ij in

· · ·· · ·

yj

i1 ij ij+1 in

· · ·· · ·

ψj






i1 i2 in

· · ·

ei

i1 ij in

· · ·· · ·

yj

i1 ij ij+1 in

· · ·· · ·

ψj






i1 i2 in

· · ·

ei

i1 ij in

· · ·· · ·

yj

i1 ij ij+1 in

· · ·· · ·

ψj





i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i i i

= 0 and

i j

= Some stuff

ki j

=

ki j

unless i = k 6= j

ii j

=

ii j

+ Plus some stuff



MV polytopes from KLR algebras

Theorem (T.-Webster, building on Kleshchev-Ram)

Fix b ∈ B(∞), and let Lb be the corresponding simple.The MV polytope MVb is the convex hull of the weights γ such that

R(γ)× R(ν − γ) ⊂ R(ν)

acts non-trivially on Lb.






R(γ)× R(ν − γ) ⊂ R(ν)






Fix b ∈ B(∞), and let Lb be the corresponding simple.

The MV polytope MVb is the convex hull of the weights γ such that

R(γ)× R(ν − γ) ⊂ R(ν)







R(γ)× R(ν − γ) ⊂ R(ν)



Three constructions of MV polytopes Sketch of a proof

Sketch of proof (not historical)

Theorem (Idea can be found in Baumann-Kamnitzer)

There is a unique map b→ Pb from B(∞) to polytopes such that, for anyb ∈ B(∞) and convex order αi = β1 ≺ β2 ≺ · · · ≺ βN = αj,

1 All edges are multiples of roots and Pb+ is a point.

2 a≺αj(Pfjb) = a≺αj

(Pb)− 1, and for all other roots a≺α (Pfjb) = a≺α (Pb).

3 If a≺αj(Pb) = 0, then for all α, a≺α (Pb) = a≺

′

sj(α)(Pσj(b)).

This map takes each b ∈ B(∞) to its MV polytope MVb.

There is a unique path through any polytope satisfying (1) that followsedge parallel to roots in the specified order; a≺αj

(Pb) is the length of theedge parallel to α in that path.σi(b) := (e∗i )maxf N

i (b) for large N is Saito’s reflection on B(∞).e∗i is Kashiwara’s ∗-crystal operator.≺′ is the order αj ≺′ sj(β1) ≺′ · · · ≺′ sj(βN−1).










′

sj(α)(Pσj(b)).














′

sj(α)(Pσj(b)).














′

sj(α)(Pσj(b)).














′

sj(α)(Pσj(b)).














′

sj(α)(Pσj(b)).














′

sj(α)(Pσj(b)).














′

sj(α)(Pσj(b)).



(Pb) is the length of theedge parallel to α in that path.

σi(b) := (e∗i )maxf Ni (b) for large N is Saito’s reflection on B(∞).

e∗i is Kashiwara’s ∗-crystal operator.≺′ is the order αj ≺′ sj(β1) ≺′ · · · ≺′ sj(βN−1).










′

sj(α)(Pσj(b)).




i (b) for large N is Saito’s reflection on B(∞).

e∗i is Kashiwara’s ∗-crystal operator.≺′ is the order αj ≺′ sj(β1) ≺′ · · · ≺′ sj(βN−1).










′

sj(α)(Pσj(b)).




i (b) for large N is Saito’s reflection on B(∞).e∗i is Kashiwara’s ∗-crystal operator.

≺′ is the order αj ≺′ sj(β1) ≺′ · · · ≺′ sj(βN−1).










′

sj(α)(Pσj(b)).






Affine MV polytopes

Affine MV polytopes

In fact, these perspectives can all be extended to the affine case

The construction in terms of Mirkovic-Vilonen cycles as of now cannot,which is one reason I have not discussed that...although there are partialresults.

One must include some extra information in the ‘polytopes,’ but thenthings like Lusztig data make sense.

The resulting objects are characterized by 2-faces (correctly interpreted).

There are really 3 steps: defining the polytopes, showing they exist, andcharacterizing 2 faces (of types A1 × A1,A2,B2,G2,A

(1)1 ,A(2)

2 ).


Affine MV polytopes

Affine MV polytopes






(1)1 ,A(2)

2 ).


Affine MV polytopes

Affine MV polytopes






(1)1 ,A(2)

2 ).


Affine MV polytopes

Affine MV polytopes






(1)1 ,A(2)

2 ).


Affine MV polytopes

Affine MV polytopes






(1)1 ,A(2)

2 ).


Affine MV polytopes

Affine MV polytopes






(1)1 ,A(2)

2 ).


Affine MV polytopes Definition

Definition of affine MV polytopes

Definition

Let g be a affine Kac-Moody algebra of rank n + 1. A type g MV polytope isa convex polytope whose edges are all parallel to roots, along with a partitionsassociated to each (possibly degenerate) vertical n-face, such that

For each edge e which is a translate of kδ, the sum of |λF| over all facesF incident to e is k.

Each 2-face is an MV polytope of the right type (sl2 × sl2, sl3, B2, G2,sl2, A(2)

2 ), where in the sl2 case one should only consider the decorationfor the n-faces F,F′ which are incident to the 2-face, but don’t contain it.

Theorem (

Baumann-Kamnitzer-T. in symmetric type. T.-Webster in generalaffine type

)

These polytopes can be used to realize B(∞) just as in finite type, and haveall the properties we want.




Definition

Let g be a affine Kac-Moody algebra of rank n + 1. A type g MV polytope isa convex polytope whose edges are all parallel to roots, along with a partitionsassociated to each (possibly degenerate) vertical n-face, such that




Theorem (


)





DefinitionLet g be a affine Kac-Moody algebra of rank n + 1.

A type g MV polytope isa convex polytope whose edges are all parallel to roots, along with a partitionsassociated to each (possibly degenerate) vertical n-face, such that




Theorem (


)





DefinitionLet g be a affine Kac-Moody algebra of rank n + 1. A type g MV polytope isa convex polytope whose edges are all parallel to roots, along with a partitionsassociated to each (possibly degenerate) vertical n-face, such that




Theorem (


)









Theorem (


)









Theorem (


)









Theorem (Baumann-Kamnitzer-T. in symmetric type. T.-Webster in generalaffine type)




Example: a vertical face of an sl3 MV polytope

(2,1)(2) (2)

5δ5δ (2)

(2)

2α0

α1 + α2α0 + δ

MV for copy of sl2 withsimple roots α0, α1 + α2.




(2,1)(2) (2)

5δ5δ (2)

(2)

2α0

α1 + α2α0 + δ





(2,1)

(2) (2)

5δ5δ (2)

(2)

2α0

α1 + α2α0 + δ





(2,1)(2)

(2)

5δ5δ (2)

(2)

2α0

α1 + α2α0 + δ





(2,1)(2) (2)

5δ5δ (2)

(2)

2α0

α1 + α2α0 + δ





(2,1)(2) (2)

5δ5δ (2)

(2)

2α0

α1 + α2α0 + δ





(2,1)(2) (2)

5δ5δ

(2)(2)

2α0

α1 + α2α0 + δ





(2,1)(2) (2)

5δ5δ

(2)(2)

2α0

α1 + α2α0 + δ





(2,1)(2) (2)

5δ5δ

(2)(2)

2α0

α1 + α2α0 + δ





(2,1)(2) (2)

5δ5δ (2)

(2)

2α0

α1 + α2α0 + δ





(2,1)(2) (2)

5δ5δ (2)

(2)

2α0

α1 + α2α0 + δ





(2,1)(2) (2)

5δ5δ (2)

(2)

2α0

α1 + α2α0 + δ



Affine MV polytopes sl2 polytopes

sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)

There is a unique decorated polytope of this type

for any choice of edge lengths on the right side.

These, along with natural crystal operators,

realize B(∞).

They have all the properties we want.

• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,

with at least one of these being an equality.

• (µk − µk−1, ω0) ≥ 0 and (µk − µk−1, ω1) ≥ 0with at least one of these being an equality.

• Either λ = λ, or λ is obtained from λ

by adding or removing a single part

of size the width of the polytope.

• λ1, λ1 are at most the width of the polytope.

•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ

•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ

•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ

4δ

8δ

2δ

δ

δ

2δ

δ

δ

•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ

4δ

8δ

2δ

δ

δ

2δ

δ

δ

•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ

•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,






• λ1, λ1 are at most the width of the polytope.•

••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ

•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ

•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ

•••

••



sl2 MV polytopes

µ1

µ2 = µ3 = · · ·

µ1 = µ2 = · · ·

µ1

= µ3 = µ2

= µ4 = µ3

µ2

µ1

Theorem (B-D-K-T)




realize B(∞).


• (µk − µk−1, ω1) ≤ 0 and (µk − µk−1, ω0) ≤ 0,







•••

••

••

•

••

3α1

α1 + δ

α0

6α1

α1 + δ

2(α0 + 2δ)

α0 + δ

α0

12δ4δ

8δ

2δ

δ

δ

2δ

δ

δ

•••

••



The crystal operators

To apply f1:

• Increase the length of the bottom right edge by 1.

• Fix the left side in the unique possible way.

• The symmetry ∗ on B(∞) is negation.

• The operators f ∗i := ∗fi∗ act at the top.

• The operators ei shrink instead of expand.

• If there is an active α0 diagonal,

e1e∗1(P) = e∗1e1(P).

• Otherwise, e1(P) = e∗1(P).

• This is essentially enough!

(must also show there is an active α0 diagonal

exactly when ϕ1((e∗1)max(P)) > ε∗1(P))

••

••

••

•

•••

• •

•

• •••• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

••

•

•••

• •

•

• •••• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

••

•

•••

• •

•

• •••• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

••

•

•••

• •

•

• •••• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

••

•

•••

• •

•

• •••• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

••

•

•••

• •

•

• •••• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

••

•

•

••• •

•

• •••• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

••

•

•

••

• •

•

• •••• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

••

•

•••

• •

•

• •••• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

••

•

•••• •

•

• •

••• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

••

•

•••• •

•

• •

••

• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

••

•

•••• •

•

• •

••

• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

••

•

•••• •

•

• •••

• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).




exactly when ϕ1((e∗1)max(P)) > ε∗1(P))••

••

••

•

•••• •

•

• •••• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

•

•••• •

•

• •••• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

•

•••• •

•

• •••• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

•

•••• •

•

• •••• •

••

•••

••

••

•

••




To apply f1:







e1e∗1(P) = e∗1e1(P).





••

••

•

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To apply f1:







e1e∗1(P) = e∗1e1(P).





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• •••• •

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To apply f1:







e1e∗1(P) = e∗1e1(P).





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•

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•

• •••• •

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To apply f1:







e1e∗1(P) = e∗1e1(P).





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•

•••• •

•

• •••

• •

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To apply f1:







e1e∗1(P) = e∗1e1(P).





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•••• •

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• •••

• •

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To apply f1:







e1e∗1(P) = e∗1e1(P).





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Affine MV polytopes Proving they exist

Proving the polytopes exist

For rank two, this can be done purely combinatorialy

It can also be done using affine PBW bases...and we expect this approachwill work in higher rank affine types.

I will explain this in terms of quiver varieties.

In fact this is not completely general, since it only works for symmetrictypes, but the KLR approach, which is general, is a bit more technical.

In fact I’ll mostly show you what happens in sl2, but will try to tell youhow the construction generalizes.











































The sl2 quiver varieties

We consider the double quiverx1, x2

y1, y2

Q = 0 1

Theorem (B-K-T)

Find the component Zb corresponding to b ∈ B(∞). Fix π ∈ Zb generic.Then the undecorated polytope corresponding to b is the convex hull of thedimension vectors of the subrepresentations of (V, π).

The decoration comes from studying finer structure of (V, π).





y1, y2

Q = 0 1

Theorem (B-K-T)







y1, y2

Q = 0 1

Theorem (B-K-T)







y1, y2

Q = 0 1

Theorem (B-K-T)





Filtrations of preprojective algebra modules

We consider various distinguished subrepresentations:· · ·· · ·

1 1 1 1

0 0 0 0AAU AAU AAU AAU��

.1

0a b .

1 1 1 1

0 0 0 0 .

· · ·· · ·AAU AAU AAU AAU��

There is a canonical filtration of (V, π) with these as subquotients (aHarder-Narasimhan (HN) filtration), where at one step one sees acomplicated extension of the second type of representation.The real edge lengths give the multiplicities in the sub-quotients.The (dual partition to) the decoration records the dimensions of theindecomposable summands at the complicated step.





1 1 1 1


.1

0a b .

1 1 1 1

0 0 0 0 .







1 1 1 1


.1

0a b .

1 1 1 1

0 0 0 0 .





Filtration for sl2 example

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1

1 1

1 10

1 summand of dim (4, 4)


6 summands of dim (1, 1)

0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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•

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1

1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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•

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1

1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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•

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1

1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1

1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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•

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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1

1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1

1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1

1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1

1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1

1 1 1 1 1




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1 1 1

1 10




0

0

0 01

0 0 01 1

0 0 01 1

1 of dim (1, 1)

1 of dim (3, 3)

1 10

1 1 1 1 1 1



Convex orders and filtrations in general

(well, for sl3)

.

......

......

...

α1

α2

α0

α1 + α2

α1 ≺ α1 + α2 ≺ α2 ≺ α1 + α2 + δ ≺ α1 + δ ≺ α1 + α2 + 2δ ≺ α2 + δ ≺ . . . δ · · · ≺ α0 + α2 ≺ α0 + α1 ≺ α0

......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0

A convex order on root directions (i.e. positive real roots and δ) is a totalordering such that the convex cones generated by any initial segment andits corresponding final segment intersect only at the origin.For every convex order, we construct a filtration of any representation Tof the preprojective algebra.The trick is to study the subquotients corresponding to δ.Even better, study the special subquotient in a nearby degenerate order.



Convex orders and filtrations in general (well, for sl3)

.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0





.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0

A convex order on root directions (i.e. positive real roots and δ) is a totalordering such that the convex cones generated by any initial segment andits corresponding final segment intersect only at the origin.

For every convex order, we construct a filtration of any representation Tof the preprojective algebra.The trick is to study the subquotients corresponding to δ.Even better, study the special subquotient in a nearby degenerate order.




.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






. ......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






. ......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0






. ......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0

A convex order on root directions (i.e. positive real roots and δ) is a totalordering such that the convex cones generated by any initial segment andits corresponding final segment intersect only at the origin.For every convex order, we construct a filtration of any representation Tof the preprojective algebra.

The trick is to study the subquotients corresponding to δ.Even better, study the special subquotient in a nearby degenerate order.




. ......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0

A convex order on root directions (i.e. positive real roots and δ) is a totalordering such that the convex cones generated by any initial segment andits corresponding final segment intersect only at the origin.For every convex order, we construct a filtration of any representation Tof the preprojective algebra.The trick is to study the subquotients corresponding to δ.

Even better, study the special subquotient in a nearby degenerate order.




. ......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0





. ......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0





.

......

......

...

α1

α2

α0

α1 + α2


......

...

α1

α1 + α2

α1

α1 + α2

α1

α1 + α2

α0




Thanks!

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Documents

icerm.brown.edu · Background Crystals Crystals (by example) 2 dim There are 6 one dimensional weight spaces and 1 two dimensional weight space. The generators F 1 and F 2 act between