Oriented Exchange Graphs & Torsion Classes

Transcription

1 1 / 25 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

2 Outline 2 / 25 1 Oriented exchange graphs 2 Torsion classes & biclosed subcategories 3 Application: maximal green sequences

3 Oriented exchange graphs Definition (Brüstle-Dupont-Pérotin) The oriented exchange graph of Q, denoted ÝÑ EGppQq, is the directed graph whose vertices are quivers mutation-equivalent to pq and whose edges are Q 1 Ñ µ k Q 1 if and only if k is green in Q 1. µ = µ Q = 1 2 µ 2 µ µ The oriented exchange graph of Q 1 Ñ 2 has maximal green sequences p1, 2q and p2, 1, 2q. 3 / 25

4 Oriented exchange graphs 4 / 25 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]

5 Torsion classes & biclosed subcategories 5 / 25 Where we are going From a quiver Q that is mutation-equivalent to 1 Ñ 2 Ñ Ñ n, one obtains a finite dimensional k-algebra Λ kq{i (known as a cluster-tilted algebra). " * representations of Λ-mod» rep k pq, Iq : Q satisfying I

6 Torsion classes & biclosed subcategories 5 / 25 Where we are going From a quiver Q that is mutation-equivalent to 1 Ñ 2 Ñ Ñ n, one obtains a finite dimensional k-algebra Λ kq{i (known as a cluster-tilted algebra). " * representations of Λ-mod» rep k pq, Iq : Q satisfying I Goal: use nice subcategories of Λ-mod to understand the poset structure of ÝÑ EGppQq.

7 Torsion classes & biclosed subcategories 5 / 25 Where we are going From a quiver Q that is mutation-equivalent to 1 Ñ 2 Ñ Ñ n, one obtains a finite dimensional k-algebra Λ kq{i (known as a cluster-tilted algebra). " * representations of Λ-mod» rep k pq, Iq : Q satisfying I Goal: use nice subcategories of Λ-mod to understand the poset structure of ÝÑ EGppQq. Theorem (Brüstle Yang, Ingalls Thomas) Let Q be mutation-equivalent to a Dynkin quiver. Then ÝÑ EGppQq torspλq as posets where Λ kq{i is the cluster-tilted algebra associated to Q.

8 Torsion classes 6 / 25 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 ÝÑ 2) The cluster-tilted algebra associated to Q is Λ kq.

9 Torsion classes 6 / 25 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 ÝÑ 2) The cluster-tilted algebra associated to Q is Λ kq. The Auslander-Reiten quiver of Λ-mod is ΓpΛ-modq ÝÑ k 0 ÝÑ 0 k k ÝÑ 0 0. k 1

10 Torsion classes 6 / 25 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 ÝÑ 2) The cluster-tilted algebra associated to Q is Λ kq. The Auslander-Reiten quiver of Λ-mod is ΓpΛ-modq ÝÑ k 0 ÝÑ 0 k k ÝÑ 0 0. We use ΓpΛ-modq to describe the torsion classes of Λ. k 1

11 Torsion classes Example (Q 1 α ÝÑ 2) torspλq = µ = µ Q = 1 2 µ 2 µ µ 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. 7 / 25

12 Torsion classes 8 / 25 The partially ordered set torspλq is a lattice (i.e. any two torsion classes T 1, T 2 P torspλq have a join (resp. meet), denoted T 1 _ T 2 (resp. T 1 ^ T 2 )).

13 Torsion classes 8 / 25 The partially ordered set torspλq is a lattice (i.e. any two torsion classes T 1, T 2 P torspλq have a join (resp. meet), denoted T 1 _ T 2 (resp. T 1 ^ T 2 )). Lemma Let Λ be a finite dimensional k-algebra and let T 1, T 2 P torspλq. Then

14 Torsion classes 8 / 25 The partially ordered set torspλq is a lattice (i.e. any two torsion classes T 1, T 2 P torspλq have a join (resp. meet), denoted T 1 _ T 2 (resp. T 1 ^ T 2 )). Lemma Let Λ be a finite dimensional k-algebra and let T 1, T 2 P torspλq. Then iq T 1 ^ T 2 T 1 X T 2, iiq T 1 _ T 2 FiltpT 1, T 2 q where FiltpT 1, T 2 q consists of objects X with a filtration 0 X 0 Ă X 1 Ă Ă X n X such that X j {X j 1 belongs to T 1 or T 2. [G. McConville]

15 Torsion classes 8 / 25 The partially ordered set torspλq is a lattice (i.e. any two torsion classes T 1, T 2 P torspλq have a join (resp. meet), denoted T 1 _ T 2 (resp. T 1 ^ T 2 )). Lemma Let Λ be a finite dimensional k-algebra and let T 1, T 2 P torspλq. Then iq T 1 ^ T 2 T 1 X T 2, iiq T 1 _ T 2 FiltpT 1, T 2 q where FiltpT 1, T 2 q consists of objects X with a filtration 0 X 0 Ă X 1 Ă Ă X n X such that X j {X j 1 belongs to T 1 or T 2. [G. McConville] Theorem (G. McConville) If Q is mutation-equivalent to a Dynkin quiver, then ÝÑ EGppQq torspλq is a semidistributive lattice (i.e. T 1 ^ T 3 T 2 ^ T 3 implies that pt 1 _ T 2 q ^ T 3 T 1 ^ T 3 and the dual statement holds).

16 Torsion classes The partially ordered set torspλq is a lattice (i.e. any two torsion classes T 1, T 2 P torspλq have a join (resp. meet), denoted T 1 _ T 2 (resp. T 1 ^ T 2 )). Lemma Let Λ be a finite dimensional k-algebra and let T 1, T 2 P torspλq. Then iq T 1 ^ T 2 T 1 X T 2, iiq T 1 _ T 2 FiltpT 1, T 2 q where FiltpT 1, T 2 q consists of objects X with a filtration 0 X 0 Ă X 1 Ă Ă X n X such that X j {X j 1 belongs to T 1 or T 2. [G. McConville] Theorem (G. McConville) If Q is mutation-equivalent to a Dynkin quiver, then ÝÑ EGppQq torspλq is a semidistributive lattice (i.e. T 1 ^ T 3 T 2 ^ T 3 implies that pt 1 _ T 2 q ^ T 3 T 1 ^ T 3 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. 8 / 25

17 Torsion classes 9 / 25 Example A lattice quotient map π Ó : L Ñ L{ is a surjective map of lattices.

18 Biclosed subcategories 10 / 25 Now we assume that Q is mutation-equivalent to 1 Ñ 2 Ñ Ñ n.

19 Biclosed subcategories 10 / 25 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 1 X iq for some set of indecomposables tx i u k i 1 (here addp k i 1 X iq consists of objects k i 1 Xm i i where m i ě 0),

20 Biclosed subcategories 10 / 25 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 1 X iq for some set of indecomposables tx i u k i 1 (here addp k i 1 X iq consists of objects k i 1 Xm i i where m i ě 0), bq B is weakly extension closed: if 0 Ñ X 1 Ñ X Ñ X 2 Ñ 0 is an exact sequence where X 1, X 2, X are indecomposable and X 1, X 2 P B, then X P B, b q B is weakly extension coclosed: if 0 Ñ X 1 Ñ X Ñ X 2 Ñ 0 " X 1, X 2 R B, then X R B.

21 Biclosed subcategories 11 / 25 Example (Q 1 α ÝÑ 2) BICpQq weak order on elements of S 3 The family of lattices of the form BICpQq properly contains the lattices isomorphic to the weak order on S n.

22 Biclosed subcategories 12 / 25 π Theorem (G. McConville) Let B addp k i 1 X iq P BICpQq and let π Ó pbq : addp l j 1X ij : X ij Y ùñ Y P Bq.

23 Biclosed subcategories 12 / 25 π Theorem (G. McConville) Let B addp k i 1 X iq P BICpQq and let π Ó pbq : addp l j 1X ij : X ij Y ùñ Y P Bq. Then π Ó : BICpQq Ñ torspλq is a lattice quotient map.

24 Biclosed subcategories 13 / 25 Theorem (G. McConville) The lattice BICpQq is semidistributive, congruence-uniform, and polygonal.

25 Biclosed subcategories 13 / 25 Theorem (G. McConville) The lattice BICpQq is semidistributive, congruence-uniform, and polygonal. Thus so is ÝÑ EGppQq 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.

26 Biclosed subcategories 13 / 25 Theorem (G. McConville) The lattice BICpQq is semidistributive, congruence-uniform, and polygonal. Thus so is ÝÑ EGppQq 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

27 Application: maximal green sequences 14 / 25 Example The maximal green sequences of Q are connected by polygonal flips.

28 Application: maximal green sequences 15 / 25 Example The maximal green sequences of Q are connected by polygonal flips.

29 Application: maximal green sequences 16 / 25 Example The maximal green sequences of Q are connected by polygonal flips.

30 Application: maximal green sequences 17 / 25 Example The maximal green sequences of Q are connected by polygonal flips.

31 Application: maximal green sequences 18 / 25 Example The maximal green sequences of Q are connected by polygonal flips.

32 Application: maximal green sequences 19 / 25 Example The maximal green sequences of Q are connected by polygonal flips.

33 Application: maximal green sequences 20 / 25 Example The maximal green sequences of Q are connected by polygonal flips.

34 Application: maximal green sequences 21 / 25 Example The maximal green sequences of Q are connected by polygonal flips.

35 Application: maximal green sequences 22 / 25 Example The maximal green sequences of Q are connected by polygonal flips.

36 Application: maximal green sequences 23 / 25 Example The maximal green sequences of Q are connected by polygonal flips.

37 24 / 25 Application: maximal green sequences Theorem (G. McConville) If Q is mutation-equivalent to 1 Ñ 2 Ñ Ñ n, ÝÑ EGppQq is a polygonal lattice whose polygons are of the form.

38 Application: maximal green sequences Theorem (G. McConville) If Q is mutation-equivalent to 1 Ñ 2 Ñ Ñ n, ÝÑ EGppQq 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 tl min, l min ` 1,..., l max 1, l max u where l min : length of the shortest maximal green sequence of Q,. l max : length of the longest maximal green sequence of Q. 24 / 25

39 Thanks! 25 / 25 π