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Agenda
- 1. Coxeter groups and Cambrian lattices
- 2. Torsion classes and stability
- 3. Relations between lattice theory and representation theory
Between lattice theory and representation theory David E Speyer - - PowerPoint PPT Presentation
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.
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Coxeter groups Let A be a symmetric n × n crystallographic Cartan matrix, meaning aii = 2 and, for i = j, we have aij = aji ∈ {0, −1, −2, · · · }. Let V be a vector space with basis α1, α2, . . . , αn and symmetric bilinear form αi · αj = aij. We write V ∨ for the dual space to V . The αi are the simple roots. Let si be the orthogonal reflection of V ∨ over α⊥
i . The si are the
simple reflections.
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Coxeter groups, continued The Coxeter group, W, is the subgroup of GL(V ∨) generated by the si. The relations of W are: s2
i
= 1 sisj = sjsi aij = 0 sisjsi = sjsisj aij = −1
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Geometry of Coxeter groups The set of real roots is Φ = W{α1, . . . , αn}. 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{α1, . . . , αn}, or negative, meaning in R≤0{α1, . . . , αn}. We write Φ = Φ+ ⊔ Φ−. Example: A = 2
−1 −1 2
- .
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Geometry of Coxeter groups, continued Contrasting example: A = 2
−2 −2 2
- .
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Geometry of Coxeter groups, continued Let D = {θ ∈ V ∨ : θ, αj ≥ 0, 1 ≤ j ≤ n} D◦ = {θ ∈ V ∨ : θ, αj > 0, 1 ≤ j ≤ n}. The wD◦ are disjoint. The wD triangulate a convex subset of V ∨ called the Tits cone. This triangulation is called the Coxeter fan.
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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
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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:
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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. s1s2s1 ◆ ◆ ◆ ◆ ♣ ♣ ♣ ♣ s2s1 s1s2 s2 ◆ ◆ ◆ ◆ ◆ ◆ s1 ♣♣♣♣♣♣ 1 . . . . . . s2s1s2 s1s2s1 s2s1 s1s2 s2 ❏ ❏ ❏ ❏ s1 tttt 1 2
−1 −1 2
- 2
−2 −2 2
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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 SpanR(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
- r final segment of R+.
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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, SpanR(R) is a 2-plane containing β, so SpanR(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.
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Shards and lattice congruences Given a lattice L, a lattice congruence is an equivalence relation on L such that, if x1 ≡ x2, then x1 ∨ y ≡ x2 ∨ y, and x1 ∧ z ≡ x2 ∧ 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
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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.
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Cambrian lattices Let Γ be the Coxeter diagram. This is the undirected graph with vertices 1, 2, . . . , n and −aij edges from i to j. Let Γ be an acyclic
- rientation 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.
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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
- n 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.
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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∈I1 si
i∈I2 si i∈I3 si
- · · · where
I1 ⊇ I2 ⊇ I3 ⊇ · · · and each product is ordered by Γ. (s1s2)(s1) ☛☛☛☛☛☛☛☛ ❑ ❑ ❑ (s1s2) s2 ■ ■ ■ ■ s1 ♣♣♣♣♣♣ 1 {α1, α1 + α2, α2} ⑥⑥⑥⑥⑥⑥⑥⑥⑥ ❚ ❚ ❚ ❚ ❚ {α1, α1 + α2} {α2} P P P P P P {α1} ❥❥❥❥❥❥❥❥❥❥ { } 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.
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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.
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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 M1 and M2 ∈ T , if there is an extension
0 → M1 → E → M2 → 0, then E ∈ T .
- For M ∈ T , if there is a surjection M ։ N, then N ∈ T .
If T1 and T2 are torsion classes, then T1 ∩ T2 is a torsion class which forms the greatest lower bound for T1 and T2. The set of all modules which can be formed by repeated extensions between elements of T1 and T2 is the least upper bound. Thus, Tors(A) is a lattice (and a complete one).
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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.
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Torsion classes and bricks First example: Let A = k
- a
⇆
b
- /ab, ba. The bricks are
S1 =
- k
- S2
=
- k
- P1
=
- k
− → k
- P2
=
- k
← − k
- Here are the torsion classes, presented as lists of bricks:
{P1, P2, S1, S2} ◗ ◗ ◗ ◗ ♠♠♠♠ {P2, S2} {P1, S1} {S2} ◗ ◗ ◗ ◗ ◗ ◗ ◗ {S1} ♠♠♠♠♠♠♠ { } This is weak order for • — •.
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Torsion classes and bricks Second example: Let A = k
- −
→ •
- . The bricks are now
S1 =
- k
- S2
=
- k
- P1
=
- k
− → k
- The torsion classes are:
{P1, S1, S2} ✆✆✆✆✆✆✆✆ ❖ ❖ ❖ ❖ {P1, S1} {S2} ❑ ❑ ❑ ❑ {S2} ♦♦♦♦♦ { } This is the Cambrian lattice for • − → •.
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Torsion classes and bricks Third example: Let A = k
- a
⇒
b
- with k = kalg. The bricks are
S1 = V1 =
- k
- V2
=
- k2
⇒ k
- a=[ 1 0 ]
b=[ 0 1 ] V3 =
- k3
⇒ k2 a=[ 1 0 0
0 1 0 ]
b=[ 0 1 0
0 0 1 ]
· · · S2 = W1 =
- k
- W2
=
- k
⇒ k2 a=[ 1
0 ]
b=[ 0
1 ]
W3 =
- k2
⇒ k3 a= 1 0
0 1 0 0
- b=
0 0
1 0 0 1
- · · ·
X(α:β) =
- k
⇒ k
- a=α
b=β for (α : β) ∈ P1
k 23
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- X(α:β),V1,V2,V3,...
W1,W2,W3,...
- ✐✐✐✐
✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵
- X(α:β),V1,V2,V3,...
W2,W3,...
- X(α:β),V1,V2,V3,...
W3,...
- .
. .
- {W1}
. . . {V1, V2} {V1} { } ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡
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Torsion classes and stability Let V = K0(A) ⊗ R and let V ∨ be the dual vector space. A basis for V is the simple modules; we’ll denote this basis α1, . . . , αn. For M ∈ Mod(A), we’ll write dim M for its class in V . Let θ ∈ V ∨. Put Tθ = {M : for all quotients M ։ N, θ, dim N ≤ 0}. Exercise: This is a torsion class. The torsion class Tθ can change when θ passes through (dim B)⊥ for a brick B. More specifically, a module B is semistable with respect to θ if θ, dim B = 0 and, for all quotients B ։ N, we have θ, dim N ≤ 0.
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Torsion classes and stability
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Torsion classes and stability
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The two main examples Path algebras Let Γ be the Coxeter diagram and let Γ be an acyclic orientation of Γ. Let k[ Γ] be the path algebra. Preprojective algebra Let DΓ be the directed graph where we replace each undirected edge with two edges
a
⇆
a∗. Let Λ[Γ] be k[DΓ]
modulo the relation
a∈(DΓ)1 aa∗ = 0.
Not included in this talk but also interesting – gentle algebras, algebras where we kill aa∗ for all a. There are lots of other interesting quotients!
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When W is finite . . . Tors(Λ[Γ]) is isomorphic to W as a lattice: Every torsion class is of the form Tθ and w → Tθ for θ ∈ wD◦. The bricks are in bijection with the shards, with each shard being the set of weights with respect to which that brick is stable. Tors(k[ Γ]) is isomorphic to Camb( Γ) as a lattice: Again, all torsion classes are of the form Tθ and correspond to facets of the Cambrian
- fan. There is precisely one brick for each β ∈ Φ+.
The maps Camb( Γ) ֒ → W and W ։ Camb( Γ) are induced functorially from the maps of algebras k[ Γ] → Λ[Γ] and Λ[Γ] → k[ Γ]. Sources: Ingalls-Thomas math/0612219 Iyama-Reading-Reiten-Thomas 1604.08401 Demonet-Iyama-Reading-Reiten-Thomas 1711.01785 Thomas 1706.00164
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When W is infinite . . . There should still be an injection W → Tors(Λ[Γ]), sending θ ∈ wD◦ to Tθ. Its image should be the torsion classes with finitely many bricks. But this only describes a portion of the torsion classes. The Cambrian lattice only describes a portion of the cluster complex.
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