Oriented Exchange Graphs & Torsion Classes Al Garver (joint - - PowerPoint PPT Presentation

Oriented Exchange Graphs & Torsion Classes

Al Garver (joint with Thomas McConville)

University of Minnesota

Representation Theory and Related Topics Seminar - Northeastern University October 30, 2015

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Outline

1

Oriented exchange graphs

2

Torsion classes & biclosed subcategories

3

Application: maximal green sequences

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Oriented exchange graphs

Definition (Brüstle-Dupont-Pérotin) The oriented exchange graph of Q, denoted Ý Ñ EGpp Qq, is the directed graph whose vertices are quivers mutation-equivalent to p Q and whose edges are Q1 Ñ µkQ1 if and only if k is green in Q1.

1 2 1′ 2′ 1 2 1′ 2′ 1 2 1′ 2′ 1 2 1′ 2′ 1 2 1′ 2′ 1 2 1′ 2′ ∼ = µ1 µ2 µ2 µ1 µ2

• Q =

The oriented exchange graph of Q “ 1 Ñ 2 has maximal green sequences p1, 2q and p2, 1, 2q.

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Oriented exchange graphs

Maximal green sequences and oriented exchange graphs have connections with Donaldson-Thomas invariants and quantum dilogarithm identities [Keller, Joyce-Song, Kontsevich-Soibelman] BPS states in string theory [Alim-Cecotti-Cordova-Espahbodi-Rastogi-Vafa] Cambrian lattices (e.g Tamari lattices) [Reading 2006]

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Torsion classes & biclosed subcategories

Where we are going From a quiver Q that is mutation-equivalent to 1 Ñ 2 Ñ ¨ ¨ ¨ Ñ n, one

• btains a finite dimensional k-algebra Λ “ kQ{I (known as a

cluster-tilted algebra). Λ-mod » repkpQ, Iq :“ " representations of Q satisfying I *

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Torsion classes & biclosed subcategories

Where we are going From a quiver Q that is mutation-equivalent to 1 Ñ 2 Ñ ¨ ¨ ¨ Ñ n, one

• btains a finite dimensional k-algebra Λ “ kQ{I (known as a

cluster-tilted algebra). Λ-mod » repkpQ, Iq :“ " representations of Q satisfying I * Goal: use nice subcategories of Λ-mod to understand the poset structure of Ý Ñ EGpp Qq.

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Torsion classes & biclosed subcategories

Where we are going From a quiver Q that is mutation-equivalent to 1 Ñ 2 Ñ ¨ ¨ ¨ Ñ n, one

• btains a finite dimensional k-algebra Λ “ kQ{I (known as a

cluster-tilted algebra). Λ-mod » repkpQ, Iq :“ " representations of Q satisfying I * Goal: use nice subcategories of Λ-mod to understand the poset structure of Ý Ñ EGpp Qq. Theorem (Brüstle–Yang, Ingalls–Thomas) Let Q be mutation-equivalent to a Dynkin quiver. Then Ý Ñ EGpp Qq – torspΛq as posets where Λ “ kQ{I is the cluster-tilted algebra associated to Q.

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Torsion classes

Theorem (Butler-Ringel) The indecomposable Λ-modules are parameterized by full, connected subquivers of Q that contain at most one arrow from each oriented cycle of Q. Example (Q “ 1

α

Ý Ñ 2) The cluster-tilted algebra associated to Q is Λ “ kQ.

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Torsion classes

Theorem (Butler-Ringel) The indecomposable Λ-modules are parameterized by full, connected subquivers of Q that contain at most one arrow from each oriented cycle of Q. Example (Q “ 1

α

Ý Ñ 2) The cluster-tilted algebra associated to Q is Λ “ kQ. The Auslander-Reiten quiver of Λ-mod is ΓpΛ-modq “ Ý Ñ k k Ý Ñ 0. k

1

Ý Ñ k

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Torsion classes

Theorem (Butler-Ringel) The indecomposable Λ-modules are parameterized by full, connected subquivers of Q that contain at most one arrow from each oriented cycle of Q. Example (Q “ 1

α

Ý Ñ 2) The cluster-tilted algebra associated to Q is Λ “ kQ. The Auslander-Reiten quiver of Λ-mod is ΓpΛ-modq “ Ý Ñ k k Ý Ñ 0. k

1

Ý Ñ k

• We use ΓpΛ-modq to describe the torsion classes of Λ.

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Torsion classes

Example (Q “ 1

α

Ý Ñ 2) torspΛq = –

1 2 1′ 2′ 1 2 1′ 2′ 1 2 1′ 2′ 1 2 1′ 2′ 1 2 1′ 2′ 1 2 1′ 2′ ∼ = µ1 µ2 µ2 µ1 µ2

• Q =

torspΛq :“ torsion classes of Λ ordered by inclusion A full, additive subcategory T of Λ-mod is a torsion class of Λ if it is aq extension closed : if X, Y P T and one has an exact sequence 0 Ñ X Ñ Z Ñ Y Ñ 0, then Z P T , bq quotient closed : X P T and X ։ Z implies Z P T .

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Torsion classes

The partially ordered set torspΛq is a lattice (i.e. any two torsion classes T1, T2 P torspΛq have a join (resp. meet), denoted T1 _ T2 (resp. T1 ^ T2)).

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Torsion classes

The partially ordered set torspΛq is a lattice (i.e. any two torsion classes T1, T2 P torspΛq have a join (resp. meet), denoted T1 _ T2 (resp. T1 ^ T2)). Lemma Let Λ be a finite dimensional k-algebra and let T1, T2 P torspΛq. Then

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Torsion classes

The partially ordered set torspΛq is a lattice (i.e. any two torsion classes T1, T2 P torspΛq have a join (resp. meet), denoted T1 _ T2 (resp. T1 ^ T2)). Lemma Let Λ be a finite dimensional k-algebra and let T1, T2 P torspΛq. Then iq T1 ^ T2 “ T1 X T2, iiq T1 _ T2 “ FiltpT1, T2q where FiltpT1, T2q consists of objects X with a filtration 0 “ X0 Ă X1 Ă ¨ ¨ ¨ Ă Xn “ X such that Xj{Xj´1 belongs to T1 or T2. [G.–McConville]

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Torsion classes

The partially ordered set torspΛq is a lattice (i.e. any two torsion classes T1, T2 P torspΛq have a join (resp. meet), denoted T1 _ T2 (resp. T1 ^ T2)). Lemma Let Λ be a finite dimensional k-algebra and let T1, T2 P torspΛq. Then iq T1 ^ T2 “ T1 X T2, iiq T1 _ T2 “ FiltpT1, T2q where FiltpT1, T2q consists of objects X with a filtration 0 “ X0 Ă X1 Ă ¨ ¨ ¨ Ă Xn “ X such that Xj{Xj´1 belongs to T1 or T2. [G.–McConville] Theorem (G.–McConville) If Q is mutation-equivalent to a Dynkin quiver, then Ý Ñ EGpp Qq – torspΛq is a semidistributive lattice (i.e. T1 ^ T3 “ T2 ^ T3 implies that pT1 _ T2q ^ T3 “ T1 ^ T3 and the dual statement holds).

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Torsion classes

The partially ordered set torspΛq is a lattice (i.e. any two torsion classes T1, T2 P torspΛq have a join (resp. meet), denoted T1 _ T2 (resp. T1 ^ T2)). Lemma Let Λ be a finite dimensional k-algebra and let T1, T2 P torspΛq. Then iq T1 ^ T2 “ T1 X T2, iiq T1 _ T2 “ FiltpT1, T2q where FiltpT1, T2q consists of objects X with a filtration 0 “ X0 Ă X1 Ă ¨ ¨ ¨ Ă Xn “ X such that Xj{Xj´1 belongs to T1 or T2. [G.–McConville] Theorem (G.–McConville) If Q is mutation-equivalent to a Dynkin quiver, then Ý Ñ EGpp Qq – torspΛq is a semidistributive lattice (i.e. T1 ^ T3 “ T2 ^ T3 implies that pT1 _ T2q ^ T3 “ T1 ^ T3 and the dual statement holds). Goal: Realize torspΛq as a quotient of a lattice with nice properties so that torspΛq will inherit these nice properties.

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Torsion classes

Example A lattice quotient map πÓ : L Ñ L{„ is a surjective map of lattices.

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Biclosed subcategories

Now we assume that Q is mutation-equivalent to 1 Ñ 2 Ñ ¨ ¨ ¨ Ñ n.

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Biclosed subcategories

Now we assume that Q is mutation-equivalent to 1 Ñ 2 Ñ ¨ ¨ ¨ Ñ n. BICpQq :“ biclosed subcategories of Λ-mod ordered by inclusion A full, additive subcategory B of Λ-mod is biclosed if aq B “ addp‘k

i“1Xiq for some set of indecomposables tXiuk i“1

(here addp‘k

i“1Xiq consists of objects ‘k i“1Xmi i

where mi ě 0),

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Biclosed subcategories

Now we assume that Q is mutation-equivalent to 1 Ñ 2 Ñ ¨ ¨ ¨ Ñ n. BICpQq :“ biclosed subcategories of Λ-mod ordered by inclusion A full, additive subcategory B of Λ-mod is biclosed if aq B “ addp‘k

i“1Xiq for some set of indecomposables tXiuk i“1

(here addp‘k

i“1Xiq consists of objects ‘k i“1Xmi i

where mi ě 0), bq B is weakly extension closed: if 0 Ñ X1 Ñ X Ñ X2 Ñ 0 is an exact sequence where X1, X2, X are indecomposable and X1, X2 P B, then X P B, b˚q B is weakly extension coclosed: if 0 Ñ X1 Ñ X Ñ X2 Ñ 0 ———————————"————————————– X1, X2 R B, then X R B.

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Biclosed subcategories

Example (Q “ 1

α

Ý Ñ 2) –

123 132 312 213 231 321

BICpQq weak order on elements of S3 The family of lattices of the form BICpQq properly contains the lattices isomorphic to the weak order on Sn.

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Biclosed subcategories

π↓

Theorem (G.– McConville) Let B “ addp‘k

i“1Xiq P BICpQq and let

πÓpBq :“ addp‘ℓ

j“1Xij : Xij ։ Y ù

ñ Y P Bq.

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Biclosed subcategories

π↓

Theorem (G.– McConville) Let B “ addp‘k

i“1Xiq P BICpQq and let

πÓpBq :“ addp‘ℓ

j“1Xij : Xij ։ Y ù

ñ Y P Bq. Then πÓ : BICpQq Ñ torspΛq is a lattice quotient map.

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Biclosed subcategories

Theorem (G.–McConville) The lattice BICpQq is semidistributive, congruence-uniform, and polygonal.

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Biclosed subcategories

Theorem (G.–McConville) The lattice BICpQq is semidistributive, congruence-uniform, and

• polygonal. Thus so is Ý

Ñ EGpp Qq – torspΛq. Theorem (Caspard–Le Conte de Poly-Barbut–Morvan 2004, Reading (proves polygonality) in forthcoming book) The weak order on the symmetric group (in fact, on any finite Coxeter group) is semidistributive, congruence-uniform, and polygonal.

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Biclosed subcategories

Theorem (G.–McConville) The lattice BICpQq is semidistributive, congruence-uniform, and

• polygonal. Thus so is Ý

Ñ EGpp Qq – torspΛq. Theorem (Caspard–Le Conte de Poly-Barbut–Morvan 2004, Reading (proves polygonality) in forthcoming book) The weak order on the symmetric group (in fact, on any finite Coxeter group) is semidistributive, congruence-uniform, and polygonal. Example Some congruence-uniform lattices

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Application: maximal green sequences

Example The maximal green sequences of Q are connected by polygonal flips.

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Application: maximal green sequences

Example The maximal green sequences of Q are connected by polygonal flips.

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Application: maximal green sequences

Example The maximal green sequences of Q are connected by polygonal flips.

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Application: maximal green sequences

Example The maximal green sequences of Q are connected by polygonal flips.

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Application: maximal green sequences

Example The maximal green sequences of Q are connected by polygonal flips.

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Application: maximal green sequences

Example The maximal green sequences of Q are connected by polygonal flips.

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Application: maximal green sequences

Example The maximal green sequences of Q are connected by polygonal flips.

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Application: maximal green sequences

Example The maximal green sequences of Q are connected by polygonal flips.

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Application: maximal green sequences

Example The maximal green sequences of Q are connected by polygonal flips.

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Application: maximal green sequences

Example The maximal green sequences of Q are connected by polygonal flips.

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Application: maximal green sequences

Theorem (G.–McConville) If Q is mutation-equivalent to 1 Ñ 2 Ñ ¨ ¨ ¨ Ñ n, Ý Ñ EGpp Qq is a polygonal lattice whose polygons are of the form .

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Application: maximal green sequences

Theorem (G.–McConville) If Q is mutation-equivalent to 1 Ñ 2 Ñ ¨ ¨ ¨ Ñ n, Ý Ñ EGpp Qq is a polygonal lattice whose polygons are of the form . Corollary (Conjectured by Brüstle-Dupont-Pérotin for any quiver Q) If Q is mutation-equivalent to 1 Ñ 2 Ñ ¨ ¨ ¨ Ñ n, the set of lengths of the maximal green sequences of Q is of the form tℓmin, ℓmin ` 1, . . . , ℓmax ´ 1, ℓmaxu where ℓmin :“ length of the shortest maximal green sequence of Q, ℓmax :“ length of the longest maximal green sequence of Q.

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Thanks!

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