Between lattice theory and representation theory David E Speyer - PowerPoint PPT Presentation

Between lattice theory and representation theory David E Speyer Based on results of Laurent Demonet, Colin Ingalls, Osamu Iyama, Nathan Reading, Idun Reiten, David E Speyer and Hugh Thomas. 1

Agenda 1. Coxeter groups and Cambrian lattices 2. Torsion classes and stability 3. Relations between lattice theory and representation theory 2

Coxeter groups Let A be a symmetric n × n crystallographic Cartan matrix, meaning a ii = 2 and, for i � = j , we have a ij = a ji ∈ { 0 , − 1 , − 2 , · · · } . Let V be a vector space with basis α 1 , α 2 , . . . , α n and symmetric bilinear form α i · α j = a ij . We write V ∨ for the dual space to V . The α i are the simple roots . Let s i be the orthogonal reflection of V ∨ over α ⊥ i . The s i are the simple reflections . 3

Coxeter groups, continued The Coxeter group, W , is the subgroup of GL( V ∨ ) generated by the s i . The relations of W are: s 2 = 1 i s i s j = s j s i a ij = 0 s i s j s i = s j s i s j a ij = − 1 4

Geometry of Coxeter groups The set of real roots is Φ = W { α 1 , . . . , α n } . The reflections of W are the reflections of V ∨ over the hyperplanes β ⊥ for β ∈ Φ. Each real root is either positive , meaning in R ≥ 0 { α 1 , . . . , α n } , or negative , meaning in R ≤ 0 { α 1 , . . . , α n } . We write Φ = Φ + ⊔ Φ − . � 2 − 1 � Example: A = . − 1 2 5

Geometry of Coxeter groups, continued � 2 − 2 � Contrasting example: A = . − 2 2 6

Geometry of Coxeter groups, continued Let { θ ∈ V ∨ : � θ, α j � ≥ 0 , 1 ≤ j ≤ n } D = { θ ∈ V ∨ : � θ, α j � > 0 , 1 ≤ j ≤ n } . D ◦ = The wD ◦ are disjoint. The wD triangulate a convex subset of V ∨ called the Tits cone . This triangulation is called the Coxeter fan . 7

Geometry of Coxeter groups, continued When W is finite, the Tits cone is all of V ∨ . In general, the Tits cone is θ ∈ V ∨ : # { β ∈ Φ + : � θ, β � < 0 } < ∞ � � . The wD ◦ are the regions of Tits \ � β ∈ Φ + β ⊥ . 8

Weak order in Coxeter groups For w ∈ W , the set of inversions of w is inv( w ) = { β ∈ Φ + : � wD ◦ , β � < 0 } . Put a partial order on W by containment of inversion sets. This is weak order : 9

Weak order in Coxeter groups, continued For W finite, this is a (complete) lattice, meaning that every subset S has a least upper bound � S and a greatest lower bound � S . For W infinite, this is a semi-lattice, meaning that every S which has an upper bound has a least upper bound and every nonempty S has a greatest lower bound. . . . . . . s 1 s 2 s 1 ◆ ◆ ♣ ◆ ♣ ◆ ♣ ♣ s 2 s 1 s 2 s 1 s 2 s 1 s 2 s 1 s 1 s 2 s 2 s 1 s 1 s 2 s 2 s 1 ◆ ◆ ♣♣♣♣♣♣ ◆ ◆ ◆ ◆ s 2 s 1 1 ❏ ❏ tttt ❏ ❏ 1 � 2 � 2 − 1 − 2 � � − 1 2 − 2 2 10

Rank two root subsystems A rank two root subsystem is a subset of Φ of the form R = Φ ∩ L where L ⊂ V is a two dimensional plane and Span R ( R ) = L . The positive roots in R come in a natural (up to reversal) linear order. A finite subset of Φ is of the form inv( w ) if and only if, for each rank two subsystem R , the intersection X ∩ R + is either an initial or final segment of R + . 11

Shards Let β ∈ Φ + . There are only finitely many rank two subsystems R containing β where β is a middle element. For each of these rank two subsystems, Span R ( R ) is a 2-plane containing β , so Span R ( R ) ⊥ is a hyperplane in β ⊥ . If we delete these hyperplanes from β ⊥ , the connected components of the remainder are the shards of dimension β . We write X for the set of shards. 12

Shards and lattice congruences Given a lattice L , a lattice congruence is an equivalence relation on L such that, if x 1 ≡ x 2 , then x 1 ∨ y ≡ x 2 ∨ y , and x 1 ∧ z ≡ x 2 ∧ z . • • • ❅ ❅ ❅ ❅ ❅ ❅ ⑦ ⑦ ⑦ ❅ ❅ ❅ ⑦ ⑦ ⑦ ❅ ❅ ❅ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ • • • • • • • • • • • • ❅ ❅ ❅ ❅ ❅ ❅ ⑦ ⑦ ⑦ ❅ ❅ ❅ ⑦ ⑦ ⑦ ❅ ❅ ❅ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ • • • • • ❅ ❅ ❅ ❅ ❅ ⑦ ⑦ ❅ ⑦ ⑦ ❅ ❅ ⑦ ⑦ ❅ ⑦ ⑦ ❅ ❅ ⑦ ⑦ ❅ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ • • • • • • • • ❅ ❅ ❅ ❅ ❅ ⑦ ⑦ ❅ ⑦ ❅ ⑦ ❅ ⑦ ⑦ ❅ ⑦ ❅ ⑦ ❅ ⑦ ⑦ ❅ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ • • 13

Shards and lattice congruences, continued Theorem (Reading) Let W be a finite Coxeter group. There is partial order � on X such that lattice congruences on W correspond to order ideals of X . Something very similar should be true in the infinite case. 14

Cambrian lattices Let Γ be the Coxeter diagram. This is the undirected graph with vertices 1, 2, . . . , n and − a ij edges from i to j . Let � Γ be an acyclic orientation of Γ. We can use � Γ to order each rank two subsystem. The � Γ-Cambrian lattice is a sub- and quotient (semi)-lattice of W . It can be defined in many ways. 15

Cambrian lattices As a quotient Let β be a real root. We will pick out a particular shard σ of dimension β ; it is determined by the condition that, for any rank two root subsystem ( α, · · · , β, · · · , γ ), we have � α, � < 0 on σ . The lattice Camb( � Γ) is the quotient of W where we remove all shards of dimension β other than this one. Geometrically, this gives a coarsening of the Coxeter fan called the Cambrian fan . 16

Cambrian lattices As a sub-(semi)-lattice , the elements of Camb( � Γ) are the sortable elements ; those elements of W which have reduced �� � �� � �� � words of the form i ∈ I 1 s i i ∈ I 2 s i i ∈ I 3 s i · · · where I 1 ⊇ I 2 ⊇ I 3 ⊇ · · · and each product is ordered by � Γ. { α 1 , α 1 + α 2 , α 2 } ( s 1 s 2 )( s 1 ) ❚ ❚ ❚ ⑥⑥⑥⑥⑥⑥⑥⑥⑥ ❑ ❚ ❚ ❑ ☛☛☛☛☛☛☛☛ ❑ { α 1 , α 1 + α 2 } ( s 1 s 2 ) { α 2 } { α 1 } s 2 s 1 P ❥❥❥❥❥❥❥❥❥❥ ■ P ♣♣♣♣♣♣ ■ P ■ P ■ P P 1 { } Sortable elements can also be described by pattern avoidance : An element w is � Γ-sortable if and only if, for each rank-two root subsystem R , the intersection inv( w ) ∩ R + is either an initial segment or the final element. 17

Cambrian lattices Theorem (Reading-S.) The Cambrian fan is the intersection of the g -vector fan with the Tits cone. References: See Reading math/0402063 , math/0507186 , math/0512339 and math/0606201 for the finite type case and Reading-S. arXiv:0803.2722 and arXiv:1111.2652 for the general case. 18

Torsion classes Let A be a k -algebra and let Mod( A ) be the collection of A -modules which are finite dimensional over k . A subset T of Mod( A ) is called a torsion class if • 0 ∈ T . • For M 1 and M 2 ∈ T , if there is an extension 0 → M 1 → E → M 2 → 0, then E ∈ T . • For M ∈ T , if there is a surjection M ։ N , then N ∈ T . If T 1 and T 2 are torsion classes, then T 1 ∩ T 2 is a torsion class which forms the greatest lower bound for T 1 and T 2 . The set of all modules which can be formed by repeated extensions between elements of T 1 and T 2 is the least upper bound. Thus, Tors( A ) is a lattice (and a complete one). 19

Torsion classes and bricks A torsion class contains many modules, but the key ones are the bricks. A module B is a brick if End( B ) is a division ring. If k is algebraically closed, it is equivalent to ask that End( B ) = k . Exercise A torsion class is determined by the set of bricks it contains. 20

Torsion classes and bricks a � � First example: Let A = k • • / � ab, ba � . The bricks are ⇆ b � � � � S 1 = k 0 S 2 = 0 k � � � � P 1 = k − → k P 2 = k ← − k Here are the torsion classes, presented as lists of bricks: { P 1 , P 2 , S 1 , S 2 } ◗ ♠♠♠♠ ◗ ◗ ◗ { P 2 , S 2 } { P 1 , S 1 } { S 2 } { S 1 } ◗ ◗ ♠♠♠♠♠♠♠ ◗ ◗ ◗ ◗ ◗ { } This is weak order for • — • . 21

Torsion classes and bricks � � Second example: Let A = k • − → • . The bricks are now � � � � S 1 = k 0 S 2 = 0 k � � P 1 = k − → k The torsion classes are: { P 1 , S 1 , S 2 } ❖ ❖ ❖ ✆✆✆✆✆✆✆✆ ❖ { P 1 , S 1 } { S 2 } { S 2 } ❑ ♦♦♦♦♦ ❑ ❑ ❑ { } This is the Cambrian lattice for • − → • . 22

Torsion classes and bricks a � � with k = k alg . The bricks are Third example: Let A = k • • ⇒ b � � S 1 = V 1 = k 0 � � k 2 a =[ 1 0 ] b =[ 0 1 ] V 2 = k ⇒ � k 2 � k 3 a =[ 1 0 0 b =[ 0 1 0 V 3 = 0 1 0 ] 0 0 1 ] · · · ⇒ � � S 2 = W 1 = 0 k � k 2 � a =[ 1 b =[ 0 W 2 = k 0 ] 1 ] ⇒ � 1 0 � 0 0 � k 3 � � � k 2 W 3 = a = b = · · · ⇒ 0 1 1 0 0 0 0 1 � � b = β for ( α : β ) ∈ P 1 X ( α : β ) = k k a = α ⇒ k 23