The crystal structure on Mirkovic-Vilonen polytopes …...The crystal structure on Mirkovi c-Vilonen...

The crystal structure on Mirkovic-Vilonen polytopes and representations of preprojective algebras

The crystal structure on Mirkovic-Vilonenpolytopes and representations of preprojective

algebras

Yong Jiang

Fakultat fur Mathematik, Universitat Bielefeld

ICRA 2012 Bielefeld; August 14, 2012

Mirkovic-Vilonen polytopes

Notations

g: complex semisimple Lie algebra

I = {1, 2, . . . , n}, n = the rank of g

h: Cartan subalgebra of g

{αi}i∈I : the simple roots, {hi}i∈I : the simple coroots

{$i}i∈I : the fundamental weights

hR =⊕

i∈I RhiW : the Weyl group

{si}i∈I : simple reflections

Γ = {w$i|w ∈W, i ∈ I}: the set of chamber weights

Notations

I = {1, 2, . . . , n}, n = the rank of g

hR =⊕

Notations

I = {1, 2, . . . , n}, n = the rank of g

hR =⊕

Notations

I = {1, 2, . . . , n}, n = the rank of g

hR =⊕

Notations

I = {1, 2, . . . , n}, n = the rank of g

hR =⊕

Notations

I = {1, 2, . . . , n}, n = the rank of g

hR =⊕

Notations

I = {1, 2, . . . , n}, n = the rank of g

hR =⊕

i∈I Rhi

W : the Weyl group

Notations

I = {1, 2, . . . , n}, n = the rank of g

hR =⊕

Notations

I = {1, 2, . . . , n}, n = the rank of g

hR =⊕

Notations

I = {1, 2, . . . , n}, n = the rank of g

hR =⊕

Definitions and examples

Pseudo-Weyl polytopes

M• = (Mγ)γ∈Γ a collection of integers

We say that M• satisfies the edge inequalities if

Mw$i +Mwsi$i +∑j 6=i

cjiMw$j ≤ 0,

for all i ∈ I and w ∈W .

Definition

If M• satisfy the edge inequalities, we can define a polytope

P (M•) = {α ∈ hR|〈α, γ〉 ≥Mγ , for all γ ∈ Γ}

associated to M•, called a pseudo-Weyl polytope.

cjiMw$j ≤ 0,

Definition

cjiMw$j ≤ 0,

Definition

cjiMw$j ≤ 0,

Definition

Pseudo-Weyl polytopes - continued

We have a map W � {vertices of P (M•)}, w 7→ µw suchthat

〈µw, w$i〉 = Mw$i , ∀i ∈ I.

The collection of coweights µ• = (µw)w∈W is called theGelfand-Goresky-MacPherson-Serganova (GGMS) datum ofthe pseudo-Weyl polytope.

P (M•) = conv(µ•).

〈µw, w$i〉 = Mw$i , ∀i ∈ I.

P (M•) = conv(µ•).

〈µw, w$i〉 = Mw$i , ∀i ∈ I.

P (M•) = conv(µ•).

〈µw, w$i〉 = Mw$i , ∀i ∈ I.

P (M•) = conv(µ•).

Pseudo-Weyl polytope - Examples

$1s1$1

s2$2s1s2$2

s2s1$1

µe µs1

µs1s2

µw0µs2s1

µs2 µe µs1

µs1s2

µw0µs2s1 = µs2

µe µs1

µs1s2

µw0µs2s1

µs2 µe µs1

µs1s2

µw0µs2s1 = µs2

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1 = µs2

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1 = µs2

µe µs1

µs1s2

µw0µs2s1

µs2 µe µs1

µs1s2

µw0µs2s1 = µs2

µe µs1

µs1s2

µw0µs2s1

µs2 µe µs1

µs1s2

µw0µs2s1 = µs2

µe µs1

µs1s2

µw0µs2s1

µs2 µe µs1

µs1s2

µw0µs2s1 = µs2

Let P = P (M•) be a pseudo-Weyl polytope.

We say that M• satisfy the Tropical Plucker relations, if foreach triple (w, i, j) such that wsi > w, wsj > w and i 6= j wehave

Mwsi$i+Mwsj$j = min(Mw$i+Mwsisj$j ,Mw$j+Mwsjsi$i).

Definition

A pseudo-Weyl polytope P (M•) is called an Mirkovic-Vilonen(MV) polytope if M• satisfy the tropical Plucker relations. And inthis case M• is called the Berenstein-Zelevinsky (BZ) datum ofP (M•).

Definition

MV polytopes - Example

Ms1$1 +Ms2$2 = min{M$1 +Ms1s2$2 ,M$2 +Ms2s1$1}.

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

MV polytopes - continued

Let P = P (M•) = conv(µ•) be an MV polytope.

The pair (µe, µw0) is called the coweight of P .

The coweight lattice P∨ acts on the set of MV polytopes bytranslation: ν + conv(µ•) = conv(µ′•), where µ′w = ν + µwfor any w ∈W .

The orbit of an MV polytope of coweight (µ1, µ2) is called astable MV polytope of coweight µ1 − µ2.

MV: the set of stable MV polytopes

The crystal structure on MV polytopes

The Lusztig datum

i = (i1, . . . , ir): a reduced word of w0

Set wik = si1 · · · sik for each 1 ≤ k ≤ r

i determines a path e = µwi0, µwi

1, . . ., µwi

r= w0 through the

1-skeleton of the MV polytope P (M•).The i-Lusztig datum of P (M•): The sequence of lengths ofthe edges n• = (n1, n2, . . . , nr) along the above path.

Theorem (Kamnitzer, 2010)

(1). For any reduced word i, taking i-Lusztig datum gives abijection MV → Nr.(2). For any reduced word i, we have a coweight-preservingbijection B ↔MV identifying the i-Lusztig datum, where B isLusztig’s canonical basis of U+

q (g∨).

Thus the set MV inherits a crystal structure from B.

The Lusztig datum

1, . . ., µwi

r= w0 through the

q (g∨).

The Lusztig datum

1, . . ., µwi

r= w0 through the

q (g∨).

The Lusztig datum

1, . . ., µwi

r= w0 through the

1-skeleton of the MV polytope P (M•).

The i-Lusztig datum of P (M•): The sequence of lengths ofthe edges n• = (n1, n2, . . . , nr) along the above path.

q (g∨).

The Lusztig datum

1, . . ., µwi

r= w0 through the

q (g∨).

The Lusztig datum

1, . . ., µwi

r= w0 through the

q (g∨).

The Lusztig datum

1, . . ., µwi

r= w0 through the

q (g∨).

The Lusztig datum

The Lusztig datum - Examples

Let P = P (M•) = conv(µ•)

Set i = (1, 2, 1), i′ = (2, 1, 2)

The i-Lusztig datum of P is(2, 1, 2)

The i′-Lusztig datum of P is(1, 2, 1)

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µs1s2

µw0µs2s1

The Lusztig datum

Set i = (1, 2, 1), i′ = (2, 1, 2)

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µs1s2

µw0µs2s1

The Lusztig datum

Set i = (1, 2, 1), i′ = (2, 1, 2)

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µs1s2

µw0µs2s1

The Lusztig datum

Set i = (1, 2, 1), i′ = (2, 1, 2)

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µs1s2

µw0µs2s1

The Lusztig datum

Set i = (1, 2, 1), i′ = (2, 1, 2)

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µs1s2

µw0µs2s1

The Kashiwara operators

For each j ∈ I, denote by fj the Kashiwara operator

Set Γj =⋃i∈IW

−j ·$i, W

−j = {w ∈W |sjw < w}.

Let P = P (M•) = conv(µ•) be an MV polytope, then fjP is theunique MV polytope whose BZ datum M ′• satisfy

M ′$j= M$j − 1, and M ′γ = Mγ , if γ ∈ Γj .

Other M ′γs are determined by the tropical Plucker relations. Thusthe description of the Kashiwara operators is non-explicit.

Set Γj =⋃i∈IW

−j ·$i, W

−j = {w ∈W |sjw < w}.

Set Γj =⋃i∈IW

−j ·$i, W

−j = {w ∈W |sjw < w}.

Set Γj =⋃i∈IW

−j ·$i, W

−j = {w ∈W |sjw < w}.

Set Γj =⋃i∈IW

−j ·$i, W

−j = {w ∈W |sjw < w}.

The Kashiwara operator - Examples

P = P (M•), where Mγ = −1,∀γ ∈ Γ.

Let us compute f1(P ) = P (M ′•).

By the theorem M ′$1= −2,

M ′$2= M ′s1$1

= M ′s1s2$2=

M ′s2s1$1= −1.

But M ′s2$2=?

Using the tropical Pluker relationM ′s1$1

+M ′s2$2= min{M ′$1

+M ′s1s2$2

,M ′$2+M ′s2s1$1

}, we candeduce that M ′s2$2

= −2.

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µ′e µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′s2

µ′e µ′s1

µ′s1s2

µ′w0µ′s2s1

P = P (M•), where Mγ = −1,∀γ ∈ Γ.

M ′$2= M ′s1$1

= M ′s1s2$2=

M ′s2s1$1= −1.

But M ′s2$2=?

+M ′s2$2= min{M ′$1

+M ′s1s2$2

,M ′$2+M ′s2s1$1

= −2.

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µ′e µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′s2

µ′e µ′s1

µ′s1s2

µ′w0µ′s2s1

P = P (M•), where Mγ = −1,∀γ ∈ Γ.

M ′$2= M ′s1$1

= M ′s1s2$2=

M ′s2s1$1= −1.

But M ′s2$2=?

+M ′s2$2= min{M ′$1

+M ′s1s2$2

,M ′$2+M ′s2s1$1

= −2.

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µ′e µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′s2

µ′e µ′s1

µ′s1s2

µ′w0µ′s2s1

P = P (M•), where Mγ = −1,∀γ ∈ Γ.

M ′$2= M ′s1$1

= M ′s1s2$2=

M ′s2s1$1= −1.

But M ′s2$2=?

+M ′s2$2= min{M ′$1

+M ′s1s2$2

,M ′$2+M ′s2s1$1

= −2.

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µ′e µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′s2

µ′e µ′s1

µ′s1s2

µ′w0µ′s2s1

P = P (M•), where Mγ = −1,∀γ ∈ Γ.

M ′$2= M ′s1$1

= M ′s1s2$2=

M ′s2s1$1= −1.

But M ′s2$2=?

+M ′s2$2= min{M ′$1

+M ′s1s2$2

,M ′$2+M ′s2s1$1

= −2.

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µ′e µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′s2

µ′e µ′s1

µ′s1s2

µ′w0µ′s2s1

P = P (M•), where Mγ = −1,∀γ ∈ Γ.

M ′$2= M ′s1$1

= M ′s1s2$2=

M ′s2s1$1= −1.

But M ′s2$2=?

+M ′s2$2= min{M ′$1

+M ′s1s2$2

,M ′$2+M ′s2s1$1

= −2.

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µ′e µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′s2

µ′e µ′s1

µ′s1s2

µ′w0µ′s2s1

The Anderson-Mirkovic operators

W−j := {w ∈W |sjw < w}, W+j := {w ∈W |sjw > w}.

Let P = P (M•) = conv(µ•) be an MV polytope. Setcj = M$j −Msj$j − 1.

Let rj : hR → hR be a map defined by h 7→ sj(h) + cjhj .

Definition

For each j ∈ I, define a polytope AMj(P ) = conv(µ′•) = P ′ to bethe smallest pseudo-Weyl polytope such that(i) µ′w = µw for all w ∈W−j ,(ii) µ′e = µe − hj ,(iii) P ′ contains µw for all w ∈W+

(iv) if w ∈W−j is such that 〈µw, hj〉 ≥ cj , then P ′ containsrj(µw).

Definition

The AM operator - Examples

P = P (M•), where Mγ = −1,∀γ ∈ Γ.

Let us computeAM1(P ) = P (M ′•).

(i) µ′w = µw for all w ∈W−1 .

(ii) µ′e = µe − h1.

(iii) P ′ contains µw for allw ∈W+

(iv) if w ∈W−1 is such that〈µw, h1〉 ≥ c1, then P ′ containsr1(µw).

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µ′s1

µ′s1s2

µ′w0

µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′e

µs2s1

µ′s1

µ′s1s2

µ′w0

µ′e

µs2s1

r1(µs2s1)

r1(µw0)

µ′s2

µ′e µ′s1

µ′s1s2

µ′w0µ′s2s1

P = P (M•), where Mγ = −1,∀γ ∈ Γ.

(ii) µ′e = µe − h1.

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µ′s1

µ′s1s2

µ′w0

µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′e

µs2s1

µ′s1

µ′s1s2

µ′w0

µ′e

µs2s1

r1(µs2s1)

r1(µw0)

µ′s2

µ′e µ′s1

µ′s1s2

µ′w0µ′s2s1

P = P (M•), where Mγ = −1,∀γ ∈ Γ.

(ii) µ′e = µe − h1.

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µ′s1

µ′s1s2

µ′w0

µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′e

µs2s1

µ′s1

µ′s1s2

µ′w0

µ′e

µs2s1

r1(µs2s1)

r1(µw0)

µ′s2

µ′e µ′s1

µ′s1s2

µ′w0µ′s2s1

P = P (M•), where Mγ = −1,∀γ ∈ Γ.

(ii) µ′e = µe − h1.

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µ′s1

µ′s1s2

µ′w0

µ′s1

µ′s1s2

µ′w0

µ′e

µ′s1

µ′s1s2

µ′w0

µ′e

µs2s1

µ′s1

µ′s1s2

µ′w0

µ′e

µs2s1

r1(µs2s1)

r1(µw0)

µ′s2

µ′e µ′s1

µ′s1s2

µ′w0µ′s2s1

P = P (M•), where Mγ = −1,∀γ ∈ Γ.

(ii) µ′e = µe − h1.

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µ′s1

µ′s1s2

µ′w0

µ′s1

µ′s1s2

µ′w0

µ′e

µ′s1

µ′s1s2

µ′w0

µ′e

µs2s1

µ′s1

µ′s1s2

µ′w0

µ′e

µs2s1

r1(µs2s1)

r1(µw0)

µ′s2

µ′e µ′s1

µ′s1s2

µ′w0µ′s2s1

P = P (M•), where Mγ = −1,∀γ ∈ Γ.

(ii) µ′e = µe − h1.

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µ′s1

µ′s1s2

µ′w0

µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′e

µs2s1

µ′s1

µ′s1s2

µ′w0

µ′e

µs2s1

r1(µs2s1)

r1(µw0)

µ′s2

µ′e µ′s1

µ′s1s2

µ′w0µ′s2s1

P = P (M•), where Mγ = −1,∀γ ∈ Γ.

(ii) µ′e = µe − h1.

µe µs1

µs1s2

µw0µs2s1

µe µs1

µs1s2

µw0µs2s1

µ′s1

µ′s1s2

µ′w0

µ′s1

µ′s1s2

µ′w0

µ′e µ′s1

µ′s1s2

µ′w0

µ′e

µs2s1

µ′s1

µ′s1s2

µ′w0

µ′e

µs2s1

r1(µs2s1)

r1(µw0)

µ′s2

µ′e µ′s1

µ′s1s2

µ′w0µ′s2s1

The Anderson-Mirkovic conjecture

Conjecture (Anderson-Mirkovic)

For any MV polytope P and any j ∈ I, AMj(P ) is an MV

polytope and AMj(P ) = fj(P ).

Type A - proved by Kamnitzer (2007)

A counterexample in type C2 was given by Kamnitzer (2007)

Type B and C - a modified version proved by Naito andSagaki (2008)

Main results

The aim is to give an explicit description of the crystalstructure on MV.

From now on, we assume that g is of type ADE.

Theorem

For any MV polytope P and any j ∈ I, we haveAMj(P ) ⊆ fj(P ). Namely fj(P ) always satisfy the conditions(i)-(iv) in the definition of AM operators.

This answers an open question of Kamnitzer in “The crystalstructure on the set of Mirkovic-Vilonen polytopes” (Adv. Math.215 (2007), 66-93).

Main results

Theorem

Main results

Theorem

Main results

Theorem

Main results

Theorem

Main results

Results - continued

Theorem

(1) Let P be an MV polytope and j ∈ I, ifεj(P ) + ε∗j (P )− 〈wt(P ), αj〉 = 0, then AMj(P ) = fj(P ), namelythe AM conjecture holds.(2) In this case, assume that P = P (M•) and AMj(P ) = P (M ′•),we have

M ′γ =

{Mγ , if γ ∈ Γj

Msjγ + cj〈hj , γ〉, if γ ∈ Γ \ Γj

So in this case we have a very explicit description of the action ofKashiwara operators.

Main results

Results - continued

Theorem

(1) Let P be an MV polytope and j ∈ I, ifεj(P ) + ε∗j (P )− 〈wt(P ), αj〉 = 0, then AMj(P ) = fj(P ), namelythe AM conjecture holds.(2) In this case, assume that P = P (M•) and AMj(P ) = P (M ′•),we have

M ′γ =

{Mγ , if γ ∈ Γj

Msjγ + cj〈hj , γ〉, if γ ∈ Γ \ Γj

So in this case we have a very explicit description of the action ofKashiwara operators.

Main results

Results - continued

Propostion

Let P be an MV polytope and j ∈ I such that ej(P ) = 0. Thenthere exists a positive integer N = N(j, P ) such that for anyn ≥ N , the assumption in the Theorem (1) holds for fnj P . In

particular, AMj(fnj P ) = fn+1

Main results

Results - continued

Propostion

Let P be an MV polytope and j ∈ I such that ej(P ) = 0. Thenthere exists a positive integer N = N(j, P ) such that for anyn ≥ N , the assumption in the Theorem (1) holds for fnj P . In

particular, AMj(fnj P ) = fn+1

Idea of the proof

MV polytopes and preprojective algebras

Λ: the preprojective algebra.

For a chamber weight γ = w$i (i ∈ I), we have a unique (upto isomorphism) Λ-submodule of Ii with dimension vector$i − γ, denoted by N(γ).

Dγ := dim Hom(N(γ),−) is a constructible function on thevariety Λv for any v. Hence it takes a constant value on adense open subset of each irreducible component Z ∈ Irr Λv.

Theorem (Baumann-Kamnitzer, 2012)

(1). For any v ∈ Nn and a irreducible component Z ∈ Irr Λv,(Dγ(Z))γ∈Γ is a BZ-datum.(2). The map

∐v∈Nn Irr Λv →MV given by

Z 7→ P ((Dγ(Z))γ∈Γ) is a crystal isomorphism.

Idea of the proof

sketch

Idea of the proof

Observation: If M• satisfy the following condition

M ′γ = min(Mγ ,Msjγ + cj〈hj , γ〉),

for all γ ∈ Γ \ Γj , then fj(P ) ⊇ AMj(P ).

Use the homological properties in the module categorymod(Λ).

We also have a new proof for the AM conjecture in type A.

Idea of the proof

sketch

Idea of the proof

sketch

Idea of the proof

sketch

Idea of the proof

Thank You!

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