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 1 / 25
Outline Oriented exchange graphs 1 Torsion classes & biclosed subcategories 2 Application: maximal green sequences 3 2 / 25
Oriented exchange graphs Definition (Brüstle-Dupont-Pérotin) The oriented exchange graph of Q , denoted Ý Ñ EG p p Q q , is the directed graph whose vertices are quivers mutation-equivalent to p Q and whose edges are Q 1 Ñ µ k Q 1 if and only if k is green in Q 1 . 1 ′ 2 ′ 1 ′ 2 ′ ∼ = 1 2 1 2 µ 2 1 ′ 2 ′ µ 2 1 2 1 ′ 2 ′ µ 1 1 2 1 ′ 2 ′ 1 2 µ 1 1 ′ 2 ′ µ 2 � Q = 1 2 The oriented exchange graph of Q “ 1 Ñ 2 has maximal green sequences p 1 , 2 q and p 2 , 1 , 2 q . 3 / 25
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] 4 / 25
Torsion classes & biclosed subcategories 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 “ k Q { I (known as a cluster-tilted algebra ). " representations of * Λ -mod » rep k p Q , I q : “ Q satisfying I 5 / 25
Torsion classes & biclosed subcategories 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 “ k Q { I (known as a cluster-tilted algebra ). " representations of * Λ -mod » rep k p Q , I q : “ Q satisfying I Goal: use nice subcategories of Λ -mod to understand the poset structure of Ý Ñ EG p p Q q . 5 / 25
Torsion classes & biclosed subcategories 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 “ k Q { I (known as a cluster-tilted algebra ). " representations of * Λ -mod » rep k p Q , I q : “ Q satisfying I Goal: use nice subcategories of Λ -mod to understand the poset structure of Ý Ñ EG p p Q q . Theorem (Brüstle–Yang, Ingalls–Thomas) Let Q be mutation-equivalent to a Dynkin quiver. Then Ý Ñ EG p p Q q – tors p Λ q as posets where Λ “ k Q { I is the cluster-tilted algebra associated to Q. 5 / 25
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. α Example ( Q “ 1 Ý Ñ 2) The cluster-tilted algebra associated to Q is Λ “ k Q . 6 / 25
�� � 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. α Example ( Q “ 1 Ý Ñ 2) The cluster-tilted algebra associated to Q is Λ “ k Q . The Auslander-Reiten quiver of Λ -mod is 1 k Ý Ñ k Γ p Λ -mod q “ � � 0 0 0 Ý Ñ k k Ý Ñ 0 . 6 / 25
� �� 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. α Example ( Q “ 1 Ý Ñ 2) The cluster-tilted algebra associated to Q is Λ “ k Q . The Auslander-Reiten quiver of Λ -mod is 1 k Ý Ñ k Γ p Λ -mod q “ � � 0 0 0 Ý Ñ k k Ý Ñ 0 . We use Γ p Λ -mod q to describe the torsion classes of Λ . 6 / 25
Torsion classes α Example ( Q “ 1 Ý Ñ 2) 1 ′ 2 ′ 1 ′ 2 ′ = ∼ 1 2 1 2 µ 2 1 ′ 2 ′ µ 2 1 2 1 ′ 2 ′ tors p Λ q = – µ 1 1 2 1 ′ 2 ′ 1 2 µ 1 1 ′ 2 ′ µ 2 � Q = 1 2 tors p Λ q : “ torsion classes of Λ ordered by inclusion A full, additive subcategory T of Λ -mod is a torsion class of Λ if it is a q 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 , b q quotient closed : X P T and X ։ Z implies Z P T . 7 / 25
Torsion classes The partially ordered set tors p Λ q is a lattice (i.e. any two torsion classes T 1 , T 2 P tors p Λ q have a join (resp. meet ), denoted T 1 _ T 2 (resp. T 1 ^ T 2 )). 8 / 25
Torsion classes The partially ordered set tors p Λ q is a lattice (i.e. any two torsion classes T 1 , T 2 P tors p Λ q 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 tors p Λ q . Then 8 / 25
Torsion classes The partially ordered set tors p Λ q is a lattice (i.e. any two torsion classes T 1 , T 2 P tors p Λ q 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 tors p Λ q . Then i q T 1 ^ T 2 “ T 1 X T 2 , ii q T 1 _ T 2 “ F ilt p T 1 , T 2 q where F ilt p T 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] 8 / 25
Torsion classes The partially ordered set tors p Λ q is a lattice (i.e. any two torsion classes T 1 , T 2 P tors p Λ q 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 tors p Λ q . Then i q T 1 ^ T 2 “ T 1 X T 2 , ii q T 1 _ T 2 “ F ilt p T 1 , T 2 q where F ilt p T 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 Ý Ñ EG p p Q q – tors p Λ q is a semidistributive lattice (i.e. T 1 ^ T 3 “ T 2 ^ T 3 implies that p T 1 _ T 2 q ^ T 3 “ T 1 ^ T 3 and the dual statement holds). 8 / 25
Torsion classes The partially ordered set tors p Λ q is a lattice (i.e. any two torsion classes T 1 , T 2 P tors p Λ q 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 tors p Λ q . Then i q T 1 ^ T 2 “ T 1 X T 2 , ii q T 1 _ T 2 “ F ilt p T 1 , T 2 q where F ilt p T 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 Ý Ñ EG p p Q q – tors p Λ q is a semidistributive lattice (i.e. T 1 ^ T 3 “ T 2 ^ T 3 implies that p T 1 _ T 2 q ^ T 3 “ T 1 ^ T 3 and the dual statement holds). Goal: Realize tors p Λ q as a quotient of a lattice with nice properties so that tors p Λ q will inherit these nice properties. 8 / 25
Torsion classes Example A lattice quotient map π Ó : L Ñ L {„ is a surjective map of lattices. 9 / 25
Biclosed subcategories Now we assume that Q is mutation-equivalent to 1 Ñ 2 Ñ ¨ ¨ ¨ Ñ n . 10 / 25
Biclosed subcategories Now we assume that Q is mutation-equivalent to 1 Ñ 2 Ñ ¨ ¨ ¨ Ñ n . BIC p Q q : “ biclosed subcategories of Λ -mod ordered by inclusion A full, additive subcategory B of Λ -mod is biclosed if B “ add p‘ k i “ 1 X i q for some set of indecomposables t X i u k a q i “ 1 i “ 1 X m i (here add p‘ k i “ 1 X i q consists of objects ‘ k where m i ě 0), i 10 / 25
Biclosed subcategories Now we assume that Q is mutation-equivalent to 1 Ñ 2 Ñ ¨ ¨ ¨ Ñ n . BIC p Q q : “ biclosed subcategories of Λ -mod ordered by inclusion A full, additive subcategory B of Λ -mod is biclosed if B “ add p‘ k i “ 1 X i q for some set of indecomposables t X i u k a q i “ 1 i “ 1 X m i (here add p‘ k i “ 1 X i q consists of objects ‘ k where m i ě 0), i b q 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 . 10 / 25
Biclosed subcategories α Example ( Q “ 1 Ý Ñ 2) 321 312 231 – 213 132 123 BIC p Q q weak order on elements of S 3 The family of lattices of the form BIC p Q q properly contains the lattices isomorphic to the weak order on S n . 11 / 25
Biclosed subcategories π ↓ Theorem (G.– McConville) Let B “ add p‘ k i “ 1 X i q P BIC p Q q and let π Ó p B q : “ add p‘ ℓ j “ 1 X i j : X i j ։ Y ù ñ Y P B q . 12 / 25
Biclosed subcategories π ↓ Theorem (G.– McConville) Let B “ add p‘ k i “ 1 X i q P BIC p Q q and let π Ó p B q : “ add p‘ ℓ j “ 1 X i j : X i j ։ Y ù ñ Y P B q . Then π Ó : BIC p Q q Ñ tors p Λ q is a lattice quotient map. 12 / 25
Biclosed subcategories Theorem (G.–McConville) The lattice BIC p Q q is semidistributive, congruence-uniform, and polygonal. 13 / 25
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