Lattice Properties of Oriented Exchange Graphs

Transcription

1 1 / 31 Lattice Properties of Oriented Exchange Graphs Al Garver (joint with Thomas McConville) Positive Grassmannians: Applications to integrable systems and super Yang-Mills scattering amplitudes July 28, 2015

2 Outline 2 / 31 1 Oriented exchange graphs 2 Torsion classes 3 Biclosed subcategories 4 Application: maximal green sequences

3 Oriented exchange graphs 3 / 31 Let Q be a finite, connected quiver without loops or 2-cycles whose vertices are rns : t1, 2,..., nu. Example Q = pq qq

4 Oriented exchange graphs 3 / 31 Let Q be a finite, connected quiver without loops or 2-cycles whose vertices are rns : t1, 2,..., nu. Definition Given a quiver Q, the framed quiver (resp. coframed quiver) of Q, denoted pq (resp. qq), is formed by (i) adding a frozen vertex i 1 for each vertex i in Q (ii) adding an arrow i Ñ i 1 (resp. i Ð i 1 ) for each vertex i in Q. Example Q = pq qq

5 Oriented exchange graphs 4 / 31 Let MutppQq denote the set of quivers mutation-equivalent to pq.

6 Oriented exchange graphs 4 / 31 Let MutppQq denote the set of quivers mutation-equivalent to pq. Definition A nonfrozen vertex of i of Q P MutppQq is green (resp. red) if all arrows between frozen vertices of Q and i point away from (resp. toward) i. Example Q = pq qq

7 Oriented exchange graphs 5 / 31 Let MutppQq denote the set of quivers mutation-equivalent to pq. Definition A nonfrozen vertex of i of Q P MutppQq is green (resp. red) if all arrows between frozen vertices of Q and i point away from (resp. toward) i. Theorem (Derksen-Weyman-Zelevinsky, Sign Coherence") Each nonfrozen vertex i of Q P MutppQq is either green or red.

8 Oriented exchange graphs 5 / 31 Let MutppQq denote the set of quivers mutation-equivalent to pq. Definition A nonfrozen vertex of i of Q P MutppQq is green (resp. red) if all arrows between frozen vertices of Q and i point away from (resp. toward) i. Theorem (Derksen-Weyman-Zelevinsky, Sign Coherence") Each nonfrozen vertex i of Q P MutppQq is either green or red. Definition A maximal green sequence of Q is a sequence i pi 1,..., i k q of mutable vertices of pq where (i) for all j P rks vertex i j is green in µ ij 1 µ i1 ppqq and

9 Oriented exchange graphs 5 / 31 Let MutppQq denote the set of quivers mutation-equivalent to pq. Definition A nonfrozen vertex of i of Q P MutppQq is green (resp. red) if all arrows between frozen vertices of Q and i point away from (resp. toward) i. Theorem (Derksen-Weyman-Zelevinsky, Sign Coherence") Each nonfrozen vertex i of Q P MutppQq is either green or red. Definition A maximal green sequence of Q is a sequence i pi 1,..., i k q of mutable vertices of pq where (i) for all j P rks vertex i j is green in µ ij 1 µ i1 ppqq and (ii) all vertices in µ ik µ i1 ppqq are red.

10 Oriented exchange graphs 6 / 31 Definition (Brüstle-Dupont-Pérotin) The oriented exchange graph of Q, denoted ÝÑ EGppQq, is the directed graph with vertices the elements of MutppQq and edges Q 1 ÝÑ µ k Q 1 if and only if k is green in Q 1. µ = µ µ µ µ The oriented exchange graph of Q 1 Ñ 2 has maximal green sequences p1, 2q and p2, 1, 2q.

11 Torsion classes 7 / 31 Goal: Understand the global structure of oriented exchange graphs using representation theory.

12 Torsion classes 7 / 31 Goal: Understand the global structure of oriented exchange graphs using representation theory. Related work on the structure of oriented exchange graphs Ordered exchange graphs [Brüstle-Yang 2014]

13 Torsion classes 7 / 31 Goal: Understand the global structure of oriented exchange graphs using representation theory. Related work on the structure of oriented exchange graphs Ordered exchange graphs [Brüstle-Yang 2014] Semi-invariants for quiver representations [Igusa-Orr-Todorov-Weyman 2009], [Chindris 2011], [Brüstle-Hermes-Igusa-Todorov 2015]

14 Torsion classes 7 / 31 Goal: Understand the global structure of oriented exchange graphs using representation theory. Related work on the structure of oriented exchange graphs Ordered exchange graphs [Brüstle-Yang 2014] Semi-invariants for quiver representations [Igusa-Orr-Todorov-Weyman 2009], [Chindris 2011], [Brüstle-Hermes-Igusa-Todorov 2015] Scattering diagrams [Gross-Hacking-Keel-Kontsevich 2014], [Muller 2015]

15 Torsion classes 7 / 31 Goal: Understand the global structure of oriented exchange graphs using representation theory. Related work on the structure of oriented exchange graphs Ordered exchange graphs [Brüstle-Yang 2014] Semi-invariants for quiver representations [Igusa-Orr-Todorov-Weyman 2009], [Chindris 2011], [Brüstle-Hermes-Igusa-Todorov 2015] Scattering diagrams [Gross-Hacking-Keel-Kontsevich 2014], [Muller 2015] Cambrian lattices [Reading 2006] and frameworks for cluster algebras [Reading-Speyer 2015, 2015, 2015]

16 Torsion classes 8 / 31 Goal: Understand the global structure of oriented exchange graphs using representation theory. Related work on the structure of oriented exchange graphs Ordered exchange graphs [Brüstle-Yang 2014] Semi-invariants for quiver representations [Igusa-Orr-Todorov-Weyman 2009], [Chindris 2011], [Brüstle-Hermes-Igusa-Todorov 2015] Scattering diagrams [Gross-Hacking-Keel-Kontsevich 2014], [Muller 2015] Cambrian lattices [Reading 2006] and frameworks for cluster algebras [Reading-Speyer 2015, 2015, 2015] Our work is based on ideas developed in [Brüstle-Yang 2014] and [Reading 2006].

17 Torsion classes 9 / 31 Theorem (Brüstle-Yang) Let Q be mutation-equivalent to a Dynkin quiver. Then ÝÑ EGppQq torspλq where Λ kq{i is the cluster-tilted (or Jacobian) algebra associated to Q.

18 Torsion classes 10 / 31 Proposition (Brüstle-Yang) Let Q be mutation-equivalent to a Dynkin quiver. Then ÝÑ EGppQq torspλq where Λ kq{i is the cluster-tilted (or Jacobian) algebra associated to Q.

19 Torsion classes 10 / 31 Proposition (Brüstle-Yang) Let Q be mutation-equivalent to a Dynkin quiver. Then ÝÑ EGppQq torspλq where Λ kq{i is the cluster-tilted (or Jacobian) algebra associated to Q. torspλq : torsion classes of Λ ordered by inclusion Let Λ be a finite dimensional k-algebra. Then a full, additive subcategory T of Λ-mod is a torsion class of Λ if it is

20 Torsion classes 10 / 31 Proposition (Brüstle-Yang) Let Q be mutation-equivalent to a Dynkin quiver. Then ÝÑ EGppQq torspλq where Λ kq{i is the cluster-tilted (or Jacobian) algebra associated to Q. torspλq : torsion classes of Λ ordered by inclusion Let Λ be a finite dimensional k-algebra. Then 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.

21 Torsion classes 10 / 31 Proposition (Brüstle-Yang) Let Q be mutation-equivalent to a Dynkin quiver. Then ÝÑ EGppQq torspλq where Λ kq{i is the cluster-tilted (or Jacobian) algebra associated to Q. torspλq : torsion classes of Λ ordered by inclusion Let Λ be a finite dimensional k-algebra. Then 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 : if X P T and one has a surjection X Z, then Z P T.

22 Torsion classes 11 / 31 Example (Q 1 α ÝÑ 2) The cluster-tilted algebra associated to Q is Λ kq.

23 Torsion classes 11 / 31 Example (Q 1 α ÝÑ 2) The cluster-tilted algebra associated to Q is Λ kq. The Auslander-Reiten quiver of Λ-mod» rep k pqq is ΓpΛ-modq ÝÑ k 0 ÝÑ 0 k k ÝÑ 0 0. k 1

24 Torsion classes 11 / 31 Example (Q 1 α ÝÑ 2) The cluster-tilted algebra associated to Q is Λ kq. The Auslander-Reiten quiver of Λ-mod» rep k pqq 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

25 Torsion classes 12 / 31 Example (Q 1 α ÝÑ 2) torspλq = 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.

26 Torsion classes Example (Q 1 α ÝÑ 2) torspλq = µ = µ µ 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. 13 / 31

27 Torsion classes 14 / 31 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 )).

28 Torsion classes 14 / 31 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

29 Torsion classes 14 / 31 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]

30 Torsion classes 14 / 31 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] Goal: Realize torspλq as a quotient of a lattice with nice properties so that torspλq will inherit these nice properties.

31 Torsion classes 15 / 31 Example A lattice quotient map π Ó : L Ñ L{ is a surjective map of lattices.

32 Biclosed subcategories 16 / 31 Now we assume that Q is mutation-equivalent to 1 Ñ 2 Ñ Ñ n or is an oriented cycle.

33 Biclosed subcategories 16 / 31 Now we assume that Q is mutation-equivalent to 1 Ñ 2 Ñ Ñ n or is an oriented cycle. 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),

34 Biclosed subcategories 16 / 31 Now we assume that Q is mutation-equivalent to 1 Ñ 2 Ñ Ñ n or is an oriented cycle. 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 (i.e. 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 (i.e. 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 R B, then X R B).

35 Biclosed subcategories 17 / 31 Example (Q 1 α ÝÑ 2) BICpQq =

36 Biclosed subcategories 18 / 31 Example (Q 1 α ÝÑ 2) π Ó ÝÑ 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.

37 Biclosed subcategories Example (Q 1 α ÝÑ 2) π Ó ÝÑ 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. 18 / 31

38 Application: maximal green sequences 19 / 31 Example Let Qp3q := Then Λ kqp3q{xβα, γβ, αγy. α 2 β 1 3. γ

39 Application: maximal green sequences 20 / 31 Example The maximal green sequences of Qp3q are connected by polygonal flips.

40 Application: maximal green sequences 21 / 31 Example The maximal green sequences of Qp3q are connected by polygonal flips.

41 Application: maximal green sequences 22 / 31 Example The maximal green sequences of Qp3q are connected by polygonal flips.

42 Application: maximal green sequences 23 / 31 Example The maximal green sequences of Qp3q are connected by polygonal flips.

43 Application: maximal green sequences 24 / 31 Example The maximal green sequences of Qp3q are connected by polygonal flips.

44 Application: maximal green sequences 25 / 31 Example The maximal green sequences of Qp3q are connected by polygonal flips.

45 Application: maximal green sequences 26 / 31 Example The maximal green sequences of Qp3q are connected by polygonal flips.

46 Application: maximal green sequences 27 / 31 Example The maximal green sequences of Qp3q are connected by polygonal flips.

47 Application: maximal green sequences 28 / 31 Example The maximal green sequences of Qp3q are connected by polygonal flips.

48 Application: maximal green sequences 29 / 31 Example The maximal green sequences of Qp3q are connected by polygonal flips.

49 Application: maximal green sequences Theorem (G. McConville) If Q is mutation-equivalent to 1 Ñ 2 Ñ Ñ n or if Q is an oriented cycle, ÝÑ EGppQq is a polygonal lattice whose polygons are of the form. 30 / 31

50 Application: maximal green sequences Theorem (G. McConville) If Q is mutation-equivalent to 1 Ñ 2 Ñ Ñ n or if Q is an oriented cycle, 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 or if Q is an oriented cycle, 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. 30 / 31

51 Thanks! 31 / 31 π