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Reverse plane partitions via representations of quivers Al Garver, - - PowerPoint PPT Presentation

Jun 04, 2023 177 likes •422 views

Reverse plane partitions via representations of quivers Al Garver, UQAM University of Michigan (joint with Rebecca Patrias and Hugh Thomas) arXiv: 1812.08345 FPSAC 2019, University of Ljubljana, Slovenia July 4, 2019 1 / 23 Outline

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SLIDE 1

Reverse plane partitions via representations of quivers

Al Garver, UQAM Ñ University of Michigan (joint with Rebecca Patrias and Hugh Thomas) arXiv: 1812.08345

FPSAC 2019, University of Ljubljana, Slovenia

July 4, 2019

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SLIDE 2

Outline

minuscule posets Auslander–Reiten quivers nilpotent endomorphisms of quiver representations promotion on reverse plane partitions

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SLIDE 3

A minuscule poset is defined by choosing a simply-laced Dynkin diagram 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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SLIDE 4

4 3 2 1 5 3 2 1 4

1 1 2 3 3 4

ρ

4 3 5 2 1

PA4,3 PD5,1 PD5,4 A reverse plane partition is an order-reversing map ρ : P Ñ Zě0.

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SLIDE 5

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

ρ:PÑZě0PRPPpPq

q|ρ| “ ź

xPP

1 1 ´ 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, Sulzgruber ‘17) Analogous identities for “skew shapes” (Morales–Pak–Panova ‘15, Naruse–Okada ‘18)

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SLIDE 6

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

ρ:PÑZě0PRPPpPq

q|ρ| “ ź

xPP

1 1 ´ qrkpxq where |ρ| :“ ř

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

We will interpret this identity in terms of quiver representations.

4

  • 3
  • 2
  • 1

k

» – 1 fi fl

  • k

» – 0

1

fi fl

  • k2

”

1 1

ı

  • k

dimpVq “ 1211

Q V a quiver a representation of Q dimension vector of V

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SLIDE 7

Any quiver Q has an Auslander–Reiten quiver ΓpQq whose vertices are the isomorphism classes of indecomposable representations of Q.

4

  • 3
  • 2
  • 1

1111

  • 1110
  • 0111
  • τ
  • 1100
  • 0110
  • τ
  • 0011
  • τ
  • 1000
  • 0100
  • τ
  • 0010
  • τ
  • 0001

τ

  • 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 into τ-orbits. tvertices of Qu Ð Ñ tτ-orbitsu

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Lemma Given a Dynkin quiver Q and a minuscule vertex m, the Hasse quiver of the minscule poset PQ,m is isomorphic to the full subquiver of ΓpQq on the representations supported at m.

4

  • 3
  • 2
  • 1

1111

  • 1110
  • 0111
  • 1100
  • 0110
  • 0011
  • 1000
  • 0100
  • 0010
  • 0001

Q ΓpQq - the Auslander–Reiten quiver of Q Let CQ,m denote the category of all representations of Q, each of whose indecomposable summands is supported at m.

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SLIDE 9

4

  • V4
  • φ4

V4

  • 3
  • V3
  • φ3

V3

  • 2
  • V2
  • φ2

V2

  • 1

V1

φ1

V1

Let φ “ pφiqi P NEndpVq :“ tnilpotent endomorphisms of Vu. Each φi λi “ pλi

1 ě ¨ ¨ ¨ ě λi rq where partition λi records the sizes of

the Jordan blocks of φi. JFpφq :“ pλ1, . . . , λnq the Jordan form data of φ Theorem (G.–Patrias–Thomas, ‘18) There is a unique maximum value of JF(¨) on NEnd(V) with respect to componentwise dominance order, denoted by GenJF(V). Moreover, it is attained on a dense open subset of NEnd(V).

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Theorem (G.–Patrias–Thomas, ‘18) The objects of CQ,m are in bijection with RPPpPQ,mq via V ÞÑ ρpVq – reverse plane partition from filling the τ-orbits

  • f PQ,m with the Jordan block sizes in GenJF(V)

11100 01101 01111 11113 00111 00102 ÞÑ 5 4 3 5 1 3 4 3 2 1 V ÞÑ ρpVq Q dimpVq “ 3585 GenJFpVq “ 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 V ÞÑ ρpVq Q GenJFpVq “ pp3q, p4, 1q, p5, 3q, p5qq Corollary ÿ

ρPRPPpPq

q|ρ| “ ÿ

VPCQ,m

qdimpVq “ ź

ViPindpCQ,mq

1 1 ´ qdimpViq “ ź

xPP

1 1 ´ qrkpxq

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Algorithmic construction of ρpVq

Apply the following piecewise linear transformations “from right to left” in ΓpQq to obtain ρ from V P CQ,m. if W is a summand of V, replace ρipVq with ρi`1pWq “ maxWăU ρipUq ` multpWq, for each V1 in the τ-orbit of W with W ă V1, replace ρipV1q with ρi`1pV1q “ maxV1ăU ρipUq ` minUăV1 ρipUq ´ ρipV1q.

11113

  • 11100
  • 01111
  • 01101
  • 00111

00102

  • ‚
  • ‚
  • ‚
  • ‚

V ΓpQq

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SLIDE 13

Algorithmic construction of ρpVq

Apply the following piecewise linear transformations “from right to left” in ΓpQq to obtain ρ from V P CQ,m. if W is a summand of V, replace ρipVq with ρi`1pWq “ maxWăU ρipUq ` multpWq, for each V1 in the τ-orbit of W with W ă V1, replace ρipV1q with ρi`1pV1q “ maxV1ăU ρipUq ` minUăV1 ρipUq ´ ρipV1q.

11113

  • 11100
  • 01111
  • 01101
  • 00111

00102

  • ‚
  • 1
  • ‚
  • ‚
  • ‚

V ΓpQq

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SLIDE 14

Algorithmic construction of ρpVq

Apply the following piecewise linear transformations “from right to left” in ΓpQq to obtain ρ from V P CQ,m. if W is a summand of V, replace ρipVq with ρi`1pWq “ maxWăU ρipUq ` multpWq, for each V1 in the τ-orbit of W with W ă V1, replace ρipV1q with ρi`1pV1q “ maxV1ăU ρipUq ` minUăV1 ρipUq ´ ρipV1q.

11113

  • 11100
  • 01111
  • 01101
  • 00111

00102

  • ‚
  • 1
  • ‚
  • ‚
  • 3
  • ‚

V ΓpQq

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SLIDE 15

Algorithmic construction of ρpVq

Apply the following piecewise linear transformations “from right to left” in ΓpQq to obtain ρ from V P CQ,m. if W is a summand of V, replace ρipVq with ρi`1pWq “ maxWăU ρipUq ` multpWq, for each V1 in the τ-orbit of W with W ă V1, replace ρipV1q with ρi`1pV1q “ maxV1ăU ρipUq ` minUăV1 ρipUq ´ ρipV1q.

11113

  • 11100
  • 01111
  • 01101
  • 00111

00102

  • 2
  • ‚
  • 1
  • ‚
  • ‚
  • 3
  • ‚

V ΓpQq

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SLIDE 16

Algorithmic construction of ρpVq

Apply the following piecewise linear transformations “from right to left” in ΓpQq to obtain ρ from V P CQ,m. if W is a summand of V, replace ρipVq with ρi`1pWq “ maxWăU ρipUq ` multpWq, for each V1 in the τ-orbit of W with W ă V1, replace ρipV1q with ρi`1pV1q “ maxV1ăU ρipUq ` minUăV1 ρipUq ´ ρipV1q.

11113

  • 11100
  • 01111
  • 01101
  • 00111

00102

  • 2
  • ‚
  • 4
  • 1
  • ‚
  • ‚
  • 3
  • ‚

V ΓpQq

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SLIDE 17

Algorithmic construction of ρpVq

Apply the following piecewise linear transformations “from right to left” in ΓpQq to obtain ρ from V P CQ,m. if W is a summand of V, replace ρipVq with ρi`1pWq “ maxWăU ρipUq ` multpWq, for each V1 in the τ-orbit of W with W ă V1, replace ρipV1q with ρi`1pV1q “ maxV1ăU ρipUq ` minUăV1 ρipUq ´ ρipV1q.

11113

  • 11100
  • 01111
  • 01101
  • 00111

00102

  • 2
  • ‚
  • 4
  • 1
  • ‚
  • ‚
  • 2
  • ‚

V ΓpQq

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SLIDE 18

Algorithmic construction of ρpVq

Apply the following piecewise linear transformations “from right to left” in ΓpQq to obtain ρ from V P CQ,m. if W is a summand of V, replace ρipVq with ρi`1pWq “ maxWăU ρipUq ` multpWq, for each V1 in the τ-orbit of W with W ă V1, replace ρipV1q with ρi`1pV1q “ maxV1ăU ρipUq ` minUăV1 ρipUq ´ ρipV1q.

11113

  • 11100
  • 01111
  • 01101
  • 00111

00102

  • 5
  • 2
  • ‚
  • 4
  • 1
  • ‚
  • ‚
  • 2
  • ‚

V ΓpQq

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SLIDE 19

Algorithmic construction of ρpVq

Apply the following piecewise linear transformations “from right to left” in ΓpQq to obtain ρ from V P CQ,m. if W is a summand of V, replace ρipVq with ρi`1pWq “ maxWăU ρipUq ` multpWq, for each V1 in the τ-orbit of W with W ă V1, replace ρipV1q with ρi`1pV1q “ maxV1ăU ρipUq ` minUăV1 ρipUq ´ ρipV1q.

11113

  • 11100
  • 01111
  • 01101
  • 00111

00102

  • 5
  • 5
  • 3
  • ‚
  • 4
  • 1
  • ‚
  • ‚
  • 2
  • ‚

V ΓpQq

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SLIDE 20

Algorithmic construction of ρpVq

Apply the following piecewise linear transformations “from right to left” in ΓpQq to obtain ρ from V P CQ,m. if W is a summand of V, replace ρipVq with ρi`1pWq “ maxWăU ρipUq ` multpWq, for each V1 in the τ-orbit of W with W ă V1, replace ρipV1q with ρi`1pV1q “ maxV1ăU ρipUq ` minUăV1 ρipUq ´ ρipV1q.

11113

  • 11100
  • 01111
  • 01101
  • 00111

00102

  • 5
  • 5
  • 3
  • ‚
  • 4
  • 1
  • ‚
  • ‚
  • 2
  • ‚

V ΓpQq

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SLIDE 21

Algorithmic construction of ρpVq

Apply the following piecewise linear transformations “from right to left” in ΓpQq to obtain ρ from V P CQ,m. if W is a summand of V, replace ρipVq with ρi`1pWq “ maxWăU ρipUq ` multpWq, for each V1 in the τ-orbit of W with W ă V1, replace ρipV1q with ρi`1pV1q “ maxV1ăU ρipUq ` minUăV1 ρipUq ´ ρipV1q.

11113

  • 11100
  • 01111
  • 01101
  • 00111

00102

  • 5
  • 5
  • 3
  • ‚
  • 4
  • 1
  • ‚
  • ‚
  • 3
  • ‚

V ΓpQq

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Periodicity for promotion pro “ t1t2t4t3

5 4 3 5 1 3

t3

ÞÑ 8 4 2 5 1 3

t4

ÞÑ 8 4 2 8 ´ 3 1 3

t2

ÞÑ 8 8 ´ 1 2 8 ´ 3 1 3

t1

ÞÑ 8 8 ´ 1 2 8 ´ 3 1 8 ´ 3

pro

ÞÑ 8 ´ 1 8 ´ 3 8 ´ 4 8 ´ 2 8 ´ 5 8 ´ 5

pro

ÞÑ 8 ´ 1 8 ´ 2 8 ´ 4 8 ´ 3 3

pro

ÞÑ 8 ´ 1 4 1 3 1 2

pro

ÞÑ 5 4 3 5 1 3 earlier results in type A (Grinberg–Roby ‘15, Musiker–Roby ‘18)

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Hvala!

  • 11100
  • 01111
  • 00102
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