Reverse plane partitions via representations of quivers

Al Garver, University of Michigan(joint with Rebecca Patrias and Hugh Thomas)

Saint Louis University Colloquium Series

December 5, 2019

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Outline

representations of quivers (and algebras)

nilpotent endomorphisms of quiver representations

minuscule posets

reverse plane partitions on minuscule posets

periodicity of promotion

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A quiver Q is a directed graph.

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oo1 2oo

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A representation X of Q is an assignment of finite-dimensional k-vectorspaces to each vertex and linear maps to each arrow.

X “

k2 k

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10

ı

oo

1��k

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01

ı

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Throughout this talk, k will be an algebraically closed field.

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A morphism of representations is a collection of linear maps so that “everysquare commutes”.

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e

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k

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reppQq - the category of all representations of Q (over k)

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A morphism of representations is a collection of linear maps so that “everysquare commutes”.

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k

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reppQq - the category of all representations of Q (over k)

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Why study quiver representations?

Theorem (Gabriel ‘72)For any finite-dimensional k-algebra A,

modpAq » modpkQ{Iq.

andmodpkQ{Iq » reppQ, Iq.

These ideas appeared earlier in [Thrall ‘47, Grothendieck ‘57, Gabriel ‘60].

Example

Q “ 1 2αoo

�3

γ

^^ kQ –ke1 ‘ ke2 ‘ ke3 ‘ kα‘ kβ‘kγ ‘ kγβ

Here modpkQq » reppQq.

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Any quiver Q has an Auslander–Reiten quiver ΓpQq whose vertices are theisomorphism classes of indecomposable representations of Q and whosearrows are irreducible morphisms. [Auslander–Reiten ‘75]

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τoo

1000

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0100

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τoo 0010

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τoo 0001τoo

Q ΓpQq - the Auslander–Reiten quiver of Q

There is a map τ called the Auslander–Reiten translation.The Auslander–Reiten translation partitions the indecomposables intoτ -orbits.When Q is Dynkin, tvertices of Qu ÐÑ tτ -orbitsu.

Theorem (Gabriel ‘72)The isomorphism classes of indecomposable representations of a Dynkinquiver are in bijection with the positive roots of the associated root system.

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A “ kQ{I - a finite dimensional algebra

X “ ppXiqi, pfaqaq P reppQ, Iq

φ “ pφiqi - a nilpotent endomorphism of X

NEnd(X) - all nilpotent endomorphisms of X

k2

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k φ3“a //

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k

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LemmaThe space NEnd(X) is an irreducible algebraic variety.

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A “ kQ{I - a finite dimensional algebra

X “ ppXiqi, pfaqaq P reppQ, Iq

φ “ pφiqi - a nilpotent endomorphism of X

NEnd(X) - all nilpotent endomorphisms of X

k2

φ1“”

0 00 0

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φ2“0

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k2 k

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k φ3“0 //

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LemmaThe space NEnd(X) is an irreducible algebraic variety.

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For each i, φi λi “ pλi1 ě ¨ ¨ ¨ ě λi

rq where partition λi records the sizesof the Jordan blocks of φi.

JFpφq :“ pλ1, . . . , λnq the Jordan form data of φ

Example

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JFpφq “ pp1, 1q, p1q, p1qq

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For λ “ pλ1 ě ¨ ¨ ¨ ě λrq and λ1 “ pλ11 ě ¨ ¨ ¨ ě λ1r1q, one has λ ď λ1 indominance order if λ1 ` ¨ ¨ ¨ ` λ` ď λ11 ` ¨ ¨ ¨ ` λ

1` for each ` ě 1.

Theorem (G.–Patrias–Thomas, ‘18)There is a unique maximum value of JF(¨) on NEnd(X) with respect tocomponentwise dominance order, denoted by GenJF(X). It is attained on adense open subset of NEnd(X).

Example

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GenJFpXq “ pp1, 1q, p1q, p1qq

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Goal: Study the invariant GenJFpXq

Question

For which subcategories C of reppQ, Iq is it the case that any object X P Cmay be recovered from GenJF(X)? We say such a subcategory is Jordanrecoverable.

ExampleUsually GenJF(X) is not enough information to recover X. Let Q “ 1 Ð 2.

X “ k 1Ð k has GenJF(X) “ pp1q, p1qq

X1 “ k 0Ð k has GenJF(X1) “ pp1q, p1qq

Theorem (G.–Patrias–Thomas ’18)Let Q be a Dynkin quiver and m a minuscule vertex of Q. The category CQ,m

of representations of Q all of whose indecomposable summands aresupported at m is Jordan recoverable.

Moreover, we classify the objects in CQ,m in terms of the combinatorics ofthe minuscule poset associated with Q and m. 12 / 33

The minuscule posets are defined by choosing a simply-laced Dynkindiagram and a minuscule vertex m.

An 1 2 ¨ ¨ ¨ n

n

Dn 1 2 ¨ ¨ ¨ n´ 2 n´ 1

6

E6 1 2 3 4 5

7

E7 1 2 3 4 5 6

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Minuscule posets

Example (Three families of minuscule posets)

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0

0

1

1 2

3

3

4

ρ

5

3

2

1

4

4

3

52

1

PA4,3 PD5,1 PD5,4

A reverse plane partition is an order-reversing map ρ : P Ñ Zě0.The objects of CQ,m will be parameterized by reverse plane partitionsdefined on the minuscule poset associated with Q and m.

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LemmaGiven a Dynkin quiver Q and a minuscule vertex m, the Hasse quiver of theminscule poset PQ,m is isomorphic to the full subquiver of ΓpQq on therepresentations supported at m.

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0001

Q ΓpQq - the Auslander–Reiten quiver of Q

DefinitionA minuscule vertex m of a Dynkin quiver Q is one where everyindecomposable representation X of Q has dimpXmq P t0, 1u.

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Theorem (Proctor ‘84)For any minuscule poset P, the generating function for reverse planepartitions on P is

ÿ

ρ:PÑZě0PRPPpPq

q|ρ| “ź

xPP

11´ qrkpxq

where |ρ| :“ř

xPP ρpxq and rk : P Ñ Zě1 is the rank function on P.

Analogous identities for order filters of certain minuscule posets(Stanley ‘71, Hillman–Grassl ‘76, Gansner ‘81, Pak ‘01, Sulzgruber2017)

Theorem (MacMahon ‘1916)If P is a type A minuscule poset, the generating function for reverse planepartitions on P whose largest entry is at most t is

ÿ

q|ρ| “rź

i“1

sź

j“1

tź

k“1

1´ qi`j`k´1

1´ qi`j`k´2 .

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Theorem (G.–Patrias–Thomas ‘18)

The objects of CQ,m are in bijection with RPPpPQ,mq viaX ÞÑ ρ – reverse plane partition from filling the τ -orbits of

PQ,m with the Jordan block sizes in GenJF(X)

11100

01101

01111

11113

00111

00102

ÞÑ

5

4

3

5

1

3

4

3

2

1

X ÞÑ ρpXq Q

dimpXq “ 3585GenJFpXq “ pp3q, p4, 1q, p5, 3q, p5qq

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Theorem (G.–Patrias–Thomas, ‘18)

The objects of CQ,m are in bijection with RPPpPQ,mq.

11100

01101

01111

11113

00111

00102

ÞÑ

5

4

3

5

1

3

4

3

2

1

X ÞÑ ρpXq Q

GenJFpXq “ pp3q, p4, 1q, p5, 3q, p5qqCorollary

ÿ

ρPRPPpPq

q|ρ| “ÿ

XPCQ,m

qdimpXq “ź

XiPindpCQ,mq

11´ qdimpXiq

“ź

xPP

11´ qrkpxq

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Idea of the proof

5

4

3

5

1

3

ÞÑ

8

4

3

5

1

3

ÞÑ

8

4

2

5

1

3

Given x P P, one can toggle a reverse plane partition ρ as follows to producea new reverse plane partition txρ. [Berenstein–Kirillov ‘95]

txρpyq “

#

maxyăy1

ρpy1q ` miny2ăy

ρpy2q ´ ρpyq : if y “ x

ρpyq : if y ‰ x.

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Idea of the proof

Build up the representation and corresponding reverse plane partitioninductively, using toggles.

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Idea of the proof

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Idea of the proof

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::

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Idea of the proof

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Idea of the proof

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Idea of the proof

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::

00111

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Idea of the proof

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::

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01111

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::

00111

00102

::

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1

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Idea of the proof

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::

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01111

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::

00111

00102

::

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Idea of the proof

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::

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Idea of the proof

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::

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01111

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::

00111

00102

::

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Periodicity for promotion pro “ t1t2t4t3

5

4

3

5

1

3

t3ÞÑ

8

4

2

5

1

3

t4ÞÑ

8

4

2

8´ 3

1

3

t2ÞÑ

8

8´ 1

2

8´ 3

1

3

t1ÞÑ

8

8´ 1

2

8´ 3

1

8´ 3

proÞÑ

8 ´ 1

8´ 3

8´ 4

8´ 2

8´ 5

8´ 5

proÞÑ

8 ´ 1

8´ 2

8´ 4

8´ 3

0

3

proÞÑ

8 ´ 1

4

1

3

1

2

proÞÑ

5

4

3

5

1

3

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Let ti be the operation of toggling every entry of ρ P RPPpPQ,mq in τ -orbit i.Let pro “ tn ¨ ¨ ¨ t1 where arrows of Q go from larger vertices point to smallervertices.

Theorem (G.–Patrias–Thomas ‘18)

We have proh “ id where h is the Coxeter number of the root system.

Brouwer and Schrijver ‘74 proved this in type A for rpps withppxq P t0, 1u.

Fon-der-Flaass ‘92 calculated the orbit sizes in this same setting.

Further properties and generalizations worked out by Panyushev ‘09,Armstrong–Stump–Thomas ‘11, Striker–Williams ‘11, Rush–Shi ‘12.

Birational promotion (actually, birational “rowmotion”) was introducedEinstein–Propp ‘13; our promotion is the “tropicalization”.

Our theorem can be deduced from work of Grinberg–Roby ‘15 andMusiker–Roby ‘18; ours is the first uniform proof.

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Additional questions

Understand GenJFpXq for other families of algebras

Other periodicity (or integrability) results?

What is the connection to Lie theory?

What about MacMahon’s theorem?

Theorem (MacMahon ‘1916)If P is a type A minuscule poset, the generating function for reverse planepartitions on P whose largest entry is at most t is

ÿ

q|ρ| “rź

i“1

sź

j“1

tź

k“1

1´ qi`j`k´1

1´ qi`j`k´2 .

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Thanks!

11100

>>

01111

>>

00102

>>

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