Biclosed sets in representation theory Al Garver, UQAM (joint with Thomas McConville and Kaveh Mousavand) Maurice Auslander Distinguished Lectures and International Conference, WHOI April 25, 2018 1 / 18
Outline Congruence-uniform lattices L Biclosed sets Shard intersection order Ψ p L q Applications 2 / 18
A finite lattice L is congruence-uniform if it may be constructed by a sequence of interval doublings from the one element lattice. Theorem (Demonet–Iyama–Reading–Reiten–Thomas, 2017) The lattice of torsion classes of a representation finite algebra Λ , denoted tors p Λ q , is congruence-uniform. A full, additive subcategory T Ă mod p Λ q is a torsion class if the following hold: if X ։ Z and X P T , then Z P T , and if 0 Ñ X Ñ Z Ñ Y Ñ 0 and X , Y P T , then Z P T . Example Λ “ k p 1 Ð Ý 2 q 3 / 18
A finite lattice L is congruence-uniform if it may be constructed by a sequence of interval doublings from the one element lattice. mod(Λ) 2 1 , 2 1 2 0 Theorem (Demonet–Iyama–Reading–Reiten–Thomas, 2017) The lattice of torsion classes of a representation finite algebra Λ , denoted tors p Λ q , is congruence-uniform. A full, additive subcategory T Ă mod p Λ q is a torsion class if the following hold: if X ։ Z and X P T , then Z P T , and if 0 Ñ X Ñ Z Ñ Y Ñ 0 and X , Y P T , then Z P T . Example Λ “ k p 1 Ð Ý 2 q 4 / 18
� � � � � � � � � � Goal : Find other examples of congruence-uniform lattices appearing in representation theory. Assume Λ “ k Q { I is a gentle algebra . The indecomposable Λ -modules are string and band modules [Wald–Waschbusch, 1985]. A word w “ γ ǫ d d ¨ ¨ ¨ γ ǫ 1 1 with γ i P Q 1 and ǫ i P t˘ 1 u is a string in Λ if w defines an irredundant walk in Q , and w and w ´ 1 : “ γ ´ ǫ 1 ¨ ¨ ¨ γ ´ ǫ d do not contain a subpath in I . 1 d Example Sequence w “ δα ´ 1 γ ´ 1 β ´ 1 is a string in Λ “ k Q {x αβ y . ” ı » fi 1 0 1 δ – 0 � 4 � k k 2 1 fl ” ı 1 1 0 β α 4 2 3 2 k k γ 3 1 M p w q Q 1 string module Str p Λ q : “ t strings in Λ u 5 / 18
B Ă Str p Λ q is closed if w 1 , w 2 P B , w 1 α ˘ 1 w 2 P Str p Λ q for some α P Q 1 ù ñ w 1 α ˘ 1 w 2 P B Bic p Λ q : “ t biclosed sets u “ t B Ă Str p Λ q : B , Str p Λ qz B are closed u Example Str(Λ) α Λ “ k p 1 Ð Ý 2 q Str p Λ q “ t 1 , 2 , α u 1 , α 2 , α α Bic p 1 Ð Ý 2 q “ 1 2 ∅ Exercise The weak order on S n ` 1 is isomorphic to Bic p 1 Ð 2 Ð ¨ ¨ ¨ Ð n q . Theorem (Palu–Pilaud–Plamondon, 2017) If Λ is gentle and | Str p Λ q| ă 8 , then Bic p Λ q is congruence-uniform. 6 / 18
Why study biclosed sets? The poset of finite biclosed sets of positive roots is isomorphic to the weak order on the corresponding Coxeter group [Kostant, 1961]. The biclosed sets in this talk were introduced to understand the lattice structure on a lattice quotient of biclosed sets [McConville, 2015], [Garver–McConville, 2015]. A geometric analogue of our lattice quotient map had already been studied for generalized permutahedra and generalized associahedra Lattice quotient maps [Hohlweg–Lange–Thomas, 2007]. 7 / 18
� � � � α � 2 Λ “ k p 1 q{x αβ, βα y (any gentle algebra can be expressed as β k Q { I where I “ x α 1 β 1 , . . . , α k β k y ) α ˚ α q{x αβ, βα, β ˚ α ˚ , α ˚ β ˚ y Π p Λ q “ k p 1 2 β β ˚ � w in Π p Λ q mapping to Str p Λ q via p α ˚ q ˘ 1 ÞÑ α ˘ 1 u Ă Str p Λ q : “ t strings r Example String α ´ 1 α ˚ P Str p Π p Λ qq , but α ´ 1 α ˚ R Ă Str p Λ q since α ´ 1 α R Str p Λ q . Theorem (G.–McConville–Mousavand) If Λ is a gentle algebra with no ouroboros , then there is an isomorphism of posets Bic p Λ q – t T X M Λ : T P tors p Π p Λ qqu where ¨ ˛ ˝ à ‚ . M Λ : “ add M p r w q w P Ă r Str p Λ q 8 / 18
� Theorem (G.–McConville–Mousavand) If Λ is a gentle algebra with no ouroboros , then there is an isomorphism of posets Bic p Λ q – t T X M Λ : T P tors p Π p Λ qqu . We refer to the categories T X M Λ as torsion shadows and denote the lattice of torsion shadows by torshad p Λ q . A string w P Str p Λ q is an ouroboros if it starts and ends at the same vertex. Example String βα is an ouroboros in α � 2 Λ “ k p 1 q{x αβ y . β Proposition (G.–McConville–Mousavand) A gentle algebra Λ has no ouroboros if and only if every indecomposable Λ -module M is a brick (i.e., End Λ p M q is a division algebra). 9 / 18
Ñ add p À � The isomorphism is given by B ÞÝ w M p r w qq where r w P Ă r Str p Λ q and u maps to u P B via p α ˚ q ˘ 1 ÞÑ α ˘ 1 . if M p r w q ։ M p r u q , then r Example α � 2 Λ “ k p 1 q{x αβ, βα y β Str(Λ) M Λ β β β ∗ β α ∗ α α α 1 , 1 ← 2 , 1 → 2 2 , 1 ← 2 , 1 → 2 1 , 1 → 2 , 1 → 2 2 , 1 ← 2 , 1 ← 2 α α β ∗ α β β β α ∗ 1 , 1 ← 2 1 , 1 → 2 2 , 1 ← 2 2 , 1 → 2 1 , 1 → 2 1 , 1 → 2 2 , 1 ← 2 2 , 1 ← 2 1 2 1 2 0 0 Bic p Λ q torshad p Λ q 10 / 18
Congruence-uniformity is equivalent to a function λ : Cov p L q Ñ P with certain properties. Say λ is a CU-labeling of L . a a b b a b c d a c c d a b a a b b a, b, c, d a b c, d c d c d a c b d a b ∅ Ψ p L q L 11 / 18
a, b, c, d a b c, d c d c d a c b d a b ∅ Ψ p L q L L a congruence-uniform lattice, λ : Cov p L q Ñ P a CU-labeling, x P L and covers y 1 , . . . y k P L , λ Ó p x q : “ t λ p y i , x qu k i “ 1 The shard intersection order of L , denoted Ψ p L q , is the collection of sets # « k ff+ ľ ψ p x q : “ labels on covering relations in y i , x i “ 1 partially ordered by inclusion. 12 / 18
� Goal : Describe Ψ p torshad p Λ qq using the representation theory of Λ . Here the CU-labeling is given by λ : Cov p torshad p Λ qq Ý Ñ M Λ M p r p T 1 X M Λ , T 2 X M Λ q ÞÝ Ñ w q where r w P Str p Π p Λ qq is the unique string satisfying ind p T 2 X M Λ q “ ind p T 1 X M Λ q Y t M p r w qu Example α � 2 Λ “ k p 1 q{x αβ, βα y β M Λ M Λ β ∗ α β α ∗ 1 , 1 → 2 , 1 → 2 2 , 1 ← 2 , 1 ← 2 β ∗ β α ∗ α → 2 , 1 → 2 ← 2 , 1 ← 2 1 1 β ∗ α β α ∗ 1 , 1 → 2 ← 2 ← 2 1 , 1 → 2 2 , 1 2 , 1 α β ∗ β α ∗ → 2 → 2 1 ← 2 ← 2 1 1 1 1 2 1 2 ∅ 0 torshad p Λ q Ψ p torshad p Λ qq 13 / 18
� Recall that a full, additive subcategory W Ă mod p Λ q is wide if W is abelian and if 0 Ñ X Ñ Z Ñ Y Ñ 0 with X , Y P W , then Z P W . Theorem (G.–McConville–Mousavand) If Λ is a gentle algebra with no ouroboros, then there is an isomorphism Ψ p torshad p Λ qq – t W X M Λ : W P wide p Π p Λ qqu . Example α � 2 Λ “ k p 1 q{x αβ, βα y β M Λ M Λ β ∗ α β α ∗ 1 , 1 → 2 , 1 → 2 2 , 1 ← 2 , 1 ← 2 β ∗ β α ∗ α → 2 , 1 → 2 ← 2 , 1 ← 2 1 1 α β ∗ β α ∗ 1 , 1 → 2 1 , 1 → 2 2 , 1 ← 2 2 , 1 ← 2 β ∗ α β α ∗ → 2 → 2 1 ← 2 ← 2 1 1 1 1 2 1 2 ∅ 0 torshad p Λ q Ψ p torshad p Λ qq 14 / 18
Theorem (G.–McConville–Mousavand) If Λ is a gentle algebra with no ouroboros, then there is an isomorphism Ψ p torshad p Λ qq – t W X M Λ : W P wide p Π p Λ qqu . We refer to the categories W X M Λ as wide shadows and denote the lattice of wide shadows by widshad p Λ q . Idea of the proof For any M p r w q P M Λ , M p r w q is a brick. For any distinct M p r w 1 q , M p r w 2 q P λ Ó p T X M Λ q where T X M Λ P torshad p Λ q , one has Hom Π p Λ q p M p r w i q , M p r w j qq “ 0 . By a theorem of Ringel, the category filt p λ Ó p T X M Λ qq consisting of modules X with a filtration 0 “ X 0 Ă X 1 Ă ¨ ¨ ¨ Ă X k “ X such that X i { X i ´ 1 P λ Ó p T X M Λ q is wide. Show that add p‘ M p r w q : M p r w q P ψ p T X M Λ qq “ filt p λ Ó p T X M Λ qqX M Λ . 15 / 18
Applications Theorem (Marks– ˇ St’oví ˇ cek, 2015) When Λ is an algebra of finite representation type, there is a bijection between torsion classes and wide subcategories given by tors p Λ q Ý Ñ wide p Λ q ÞÝ Ñ t X P T : p g : Y Ñ X q P T , ker p g q P T u T filt p gen p W qq Ð Ý� W . Corollary There is a bijection from torsion shadows to wide shadows given by torshad p Λ q Ý Ñ widshad p Λ q T X M Λ ÞÝ Ñ add p‘ M p r w q : M p r w q P ψ p T X M Λ qq filt p gen p W qq X M Λ Ð Ý� W X M Λ . 16 / 18
Corollary The poset widshad p Λ q is a lattice. Proof. The category M Λ is the unique maximal element. The poset widshad p Λ q is closed under intersections p W 1 X M Λ q X p W 2 X M Λ q “ p W 1 X W 2 q X M Λ . The poset widshad p Λ q is finite. Problem (Reading, 2016) For which class of finite lattices L is Ψ p L q a lattice? (If L is congruence-uniform, then Ψ p L q is a partial order.) 17 / 18
Thanks! 18 / 18
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