cluster algebras and infinite associahedrafinite (real!) reflection groups (continued) theorem. a...
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![Page 1: Cluster algebras and infinite associahedraFinite (Real!) reflection groups (continued) Theorem. A finite group is a Coxeter group if and only if it is a finite reflection group](https://reader033.vdocuments.net/reader033/viewer/2022053020/5f2a595b246262549215453d/html5/thumbnails/1.jpg)
Cluster algebras and infinite associahedra
Nathan Reading
NC State University
CombinaTexas 2008
Coxeter groups
Associahedra and cluster algebras
Sortable elements/Cambrian fans
Infinite type
Much of the work described here is joint with David Speyer.
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Coxeter groups
Coxeter groups
A Coxeter group is a group with a certain presentation. Choose a
finite generating set S = {s1, . . . , sn} and for every i < j , choose
an integer m(i , j) ≥ 2. Define:
W =⟨S | s2
i = 1, ∀ i and (sisj)m(i ,j) = 1, ∀ i < j
⟩.
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Coxeter groups
Coxeter groups
A Coxeter group is a group with a certain presentation. Choose a
finite generating set S = {s1, . . . , sn} and for every i < j , choose
an integer m(i , j) ≥ 2. Define:
W =⟨S | s2
i = 1, ∀ i and (sisj)m(i ,j) = 1, ∀ i < j
⟩.
Why would anyone write this down?
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Finite (Real!) reflection groups
A hyperplane in Rd is a subspace of dimension d − 1.
An orthogonal reflection is a linear transformation which fixes ahyperplane with a (−1)-eigenspace orthogonal to the hyperplane.A finite reflection group is a finite group generated by reflections.
Example (Symmetries of a square)
s1 s2
All symmetries of the square are com-positions of the reflections s1 and s2.This is a finite reflection group.
The composition s1s2 is a 90◦ rotation,so (s1s2)
4 = 1. Abstractly, this group is
⟨{s1, s2} | s2
1 = s22 = (s1s2)
4⟩.
This is a Coxeter group.
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Finite (Real!) reflection groups (continued)
Theorem. A finite group is a Coxeter group if and only if it is afinite reflection group.
Finite reflection groups are interesting for many reasons, includingapplications to Lie theory and algebraic geometry. Finite reflectiongroups enjoy very pretty invariant theory. (Today at 1:45.)
Besides the generators S , other elements act as reflections. Thecollection A of reflecting hyperplanes for all these reflections cutsspace into “regions.” The generators S are the reflections in thewalls of some region D. Identify D with the identity element 1.The map w 7→ wD is a bijection from W to A-regions.
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Finite (Real!) reflection groups (continued)
Theorem. A finite group is a Coxeter group if and only if it is afinite reflection group.
Finite reflection groups are interesting for many reasons, includingapplications to Lie theory and algebraic geometry. Finite reflectiongroups enjoy very pretty invariant theory. (Today at 1:45.)
Besides the generators S , other elements act as reflections. Thecollection A of reflecting hyperplanes for all these reflections cutsspace into “regions.” The generators S are the reflections in thewalls of some region D. Identify D with the identity element 1.The map w 7→ wD is a bijection from W to A-regions.
s1s2 4
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Finite (Real!) reflection groups (continued)
Theorem. A finite group is a Coxeter group if and only if it is afinite reflection group.
Finite reflection groups are interesting for many reasons, includingapplications to Lie theory and algebraic geometry. Finite reflectiongroups enjoy very pretty invariant theory. (Today at 1:45.)
Besides the generators S , other elements act as reflections. Thecollection A of reflecting hyperplanes for all these reflections cutsspace into “regions.” The generators S are the reflections in thewalls of some region D. Identify D with the identity element 1.The map w 7→ wD is a bijection from W to A-regions.
s1s2 4
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Finite (Real!) reflection groups (continued)
Theorem. A finite group is a Coxeter group if and only if it is afinite reflection group.
Finite reflection groups are interesting for many reasons, includingapplications to Lie theory and algebraic geometry. Finite reflectiongroups enjoy very pretty invariant theory. (Today at 1:45.)
Besides the generators S , other elements act as reflections. Thecollection A of reflecting hyperplanes for all these reflections cutsspace into “regions.” The generators S are the reflections in thewalls of some region D. Identify D with the identity element 1.The map w 7→ wD is a bijection from W to A-regions.
s1s2 4
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Finite (Real!) reflection groups (continued)
Theorem. A finite group is a Coxeter group if and only if it is afinite reflection group.
Finite reflection groups are interesting for many reasons, includingapplications to Lie theory and algebraic geometry. Finite reflectiongroups enjoy very pretty invariant theory. (Today at 1:45.)
Besides the generators S , other elements act as reflections. Thecollection A of reflecting hyperplanes for all these reflections cutsspace into “regions.” The generators S are the reflections in thewalls of some region D. Identify D with the identity element 1.The map w 7→ wD is a bijection from W to A-regions.
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Finite (Real!) reflection groups (continued)
Theorem. A finite group is a Coxeter group if and only if it is afinite reflection group.
Finite reflection groups are interesting for many reasons, includingapplications to Lie theory and algebraic geometry. Finite reflectiongroups enjoy very pretty invariant theory. (Today at 1:45.)
Besides the generators S , other elements act as reflections. Thecollection A of reflecting hyperplanes for all these reflections cutsspace into “regions.” The generators S are the reflections in thewalls of some region D. Identify D with the identity element 1.The map w 7→ wD is a bijection from W to A-regions.
1
s1
s2 4
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Finite (Real!) reflection groups (continued)
Theorem. A finite group is a Coxeter group if and only if it is afinite reflection group.
Finite reflection groups are interesting for many reasons, includingapplications to Lie theory and algebraic geometry. Finite reflectiongroups enjoy very pretty invariant theory. (Today at 1:45.)
Besides the generators S , other elements act as reflections. Thecollection A of reflecting hyperplanes for all these reflections cutsspace into “regions.” The generators S are the reflections in thewalls of some region D. Identify D with the identity element 1.The map w 7→ wD is a bijection from W to A-regions.
1
s1
s2
s1s2
s2s1
s2s1s2
s1s2s1* ∗ = s1s2s1s2 = s2s1s2s1
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Symmetric group S4 (symmetries of regular tetrahedron)
Blue region is 1.
Regions ↔elements.
Largest circles:hyperplanes fors1, s2, and s3.(s2 on top.)
m(s1, s2) = 3.m(s2, s3) = 3.m(s1, s3) = 2.
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Symmetric group S4 (symmetries of regular tetrahedron)
Blue region is 1.
Regions ↔elements.
Largest circles:hyperplanes fors1, s2, and s3.(s2 on top.)
m(s1, s2) = 3.m(s2, s3) = 3.m(s1, s3) = 2.
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Symmetric group S4 (symmetries of regular tetrahedron)
Blue region is 1.
Regions ↔elements.
Largest circles:hyperplanes fors1, s2, and s3.(s2 on top.)
m(s1, s2) = 3.m(s2, s3) = 3.m(s1, s3) = 2.
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Symmetric group S4 (symmetries of regular tetrahedron)
Blue region is 1.
Regions ↔elements.
Largest circles:hyperplanes fors1, s2, and s3.(s2 on top.)
m(s1, s2) = 3.m(s2, s3) = 3.m(s1, s3) = 2.
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Infinite groups
Theorem. Any discrete group generated by reflections in a(constant curvature) spherical, Euclidean or hyperbolic space is aCoxeter group.
Theorem. For any Coxeter group W , one can define a symmetricbilinear form B on R
S and construct W as a discrete subgroup ofthe orthogonal group O(RS ,B) generated by reflections (i.e.reflections that are orthogonal with respect to B).
Again, a particular region (connected component of RS minus the
reflecting hyperplanes) represents the identity. Important point:The Tits cone is defined to be the union of closures of all regionsin the W -orbit of the region representing 1. When W is infinite,the Tits cone is not all of R
S .
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A reflection group in the Poincare disk
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A reflection group in the Poincare disk
1
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A reflection group in the Poincare disk
1
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A reflection group in the Poincare disk
1a
b
c
m(a, b) = 5, m(b, c) = 4, m(a, c) = 27
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A reflection group in the Euclidean plane
1
a
b
c
m(a, b) = 3, m(b, c) = 6, m(a, c) = 28
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The weak order on a Coxeter group
A reduced word for w ∈ W : a shortest possible sequence ofelements of S such that the product of the sequence is w .The weak order on W : the poset generated by relations ws < w
when s ∈ S and w has a reduced word ending in s.Weak order encodes the structure of reduced words.
Example: weakorder on Sn
Move “up” byswapping twoadjacent entries ofthe permutation, soas to put them out oforder.
S = {a, b}a = (1 2), b = (2 3)
1
a b
ab ba
aba = bab
123
213
312
132
231
321
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The weak order as a poset of regions (Example: S4)
Recall: Group elements ↔ regions (in Tits cone).Geometric characterization of weak order: go “up” from 1 bycrossing hyperplanes.
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The weak order as a poset of regions (Example: S4)
Recall: Group elements ↔ regions (in Tits cone).Geometric characterization of weak order: go “up” from 1 bycrossing hyperplanes.
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The weak order as a poset of regions (Example: S4)
Recall: Group elements ↔ regions (in Tits cone).Geometric characterization of weak order: go “up” from 1 bycrossing hyperplanes.
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The weak order as a poset of regions (Example: S4)
Recall: Group elements ↔ regions (in Tits cone).Geometric characterization of weak order: go “up” from 1 bycrossing hyperplanes.
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The weak order as a poset of regions (Example: S4)
Recall: Group elements ↔ regions (in Tits cone).Geometric characterization of weak order: go “up” from 1 bycrossing hyperplanes.
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The weak order as a poset of regions (Example: S4)
Recall: Group elements ↔ regions (in Tits cone).Geometric characterization of weak order: go “up” from 1 bycrossing hyperplanes.
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The weak order as a poset of regions (Example: S4)
Recall: Group elements ↔ regions (in Tits cone).Geometric characterization of weak order: go “up” from 1 bycrossing hyperplanes.
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The weak order as a poset of regions (Example: S4)
Recall: Group elements ↔ regions (in Tits cone).Geometric characterization of weak order: go “up” from 1 bycrossing hyperplanes.
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First take-home lesson
The beauty of Coxeter groups lies in the rich interplay betweengeometry, combinatorics and order/lattice theory.
Most proofs about Coxeter groups exploit a combination ofmethods:
1. Combinatorics of words in S .
2. Geometry of arrangements of reflecting hyperplanes.(Alternately, geometry of root systems. A root system is acollection of two normal vectors per hyperplane, chosen sothat the root system is permuted by W .)
3. Order/lattice theory of the weak order. (Alternately, theBruhat order.)
4. Linear algebra.
5. More. . .
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The classical associahedron
An n-dimensional polytope whose vertices are labeled bytriangulations of a convex (n + 3)-gon. (Counted by the Catalannumber.) Edges correspond to diagonal flips. For n = 3:
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Combinatorial clusters
Root system Φ for W : two normals for each reflecting hyperplane.Positive roots: nice choice of one root for each hyperplane.Simple roots: positive roots associated to the generators S .Almost positive roots: positive roots union negatives of simples.
Triangulations of a polygon are generalized by clusters: maximalsets of pairwise “compatible” roots in Φ≥−1.
Clusters are counted by Cat(W ), the W -Catalan number. (WhenW is the symmetric group, this is the usual Catalan number.)
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Rank-two examples (i.e. |S | = 2)
Root systems.
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Rank-two examples (i.e. |S | = 2)
Positive roots. (Simple roots in red.)
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Rank-two examples (i.e. |S | = 2)
Almost positive roots.
In these examples, two almost positive roots are compatible if andonly if they are “adjacent.” Notice that the positive linear spans ofclusters decompose space into cones. This happens in general, andthe decomposition is called the cluster fan. The cluster fan is dualto a polytope called the generalized associahedron.
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Rank-two examples (i.e. |S | = 2)
Almost positive roots.
In these examples, two almost positive roots are compatible if andonly if they are “adjacent.” Notice that the positive linear spans ofclusters decompose space into cones. This happens in general, andthe decomposition is called the cluster fan. The cluster fan is dualto a polytope called the generalized associahedron.
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Example (W = Sn)
Simple root: αi corresponding to “reflection” (i i+1).Positive root: αij = αi + αi+1 + · · ·αj .Almost positive roots ↔ diagonals of (n+2)-gon.Compatible ↔ noncrossing.Negative simple roots ↔ diagonals forming “snake.”Positive root αij ↔ diagonal crossing −αi , . . . ,−αj and no othernegative simple. (α23 shown dotted.)
−α1
−α2
−α3
−α4
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Example (W = Sn)
Simple root: αi corresponding to “reflection” (i i+1).Positive root: αij = αi + αi+1 + · · ·αj .Almost positive roots ↔ diagonals of (n+2)-gon.Compatible ↔ noncrossing.Negative simple roots ↔ diagonals forming “snake.”Positive root αij ↔ diagonal crossing −αi , . . . ,−αj and no othernegative simple. (α23 shown dotted.)
−α1
−α2
−α3
−α4Compatibility for general W : Thereis a “rotation” on Φ≥−1 such thatevery positive root can be rotated toa negative simple root. Compatibilityis rotation invariant and −α iscompatible with β is and only if α
has coefficient zero in thesimple-root expansion of β.
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Cluster algebras (Fomin and Zelevinsky)
Generalized associahedra are the underlying combinatorialstructures for cluster algebras of finite type.Initial motivation: a framework for the study of total positivity andof Lusztig/Kashiwara’s canonical bases of quantum groups. Manyalgebras related to reductive Lie groups have the structure of acluster algebra.
Subsequent applications (due to many researchers):
1. Discrete dynamical systems based on rational recurrences.
2. Y -systems in thermodynamic Bethe Ansatz.
3. Quiver representations.
4. Grassmannians, projective configurations & tropical analogues.
5. Poisson geometry, Teichmuller theory.
6. Triangulations of orientable surfaces.
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Definition of a cluster algebra (modulo all the details)
A seed consists of a cluster of n rational functions, called clustervariables and matrix B .
For each cluster variable x in a seed, mutation creates a new seed:
x is replaced with a new rational function x ′.
The other cluster variables are unchanged.
B is replaced with a new matrix.
(B is the combinatorial data telling you how to do mutations.)
Start with an initial seed and let mutations propagate in alldirections to obtain a (usually infinite) collection of seeds. Use thecollection of all cluster variables in all these seeds to generate analgebra. This is the cluster algebra associated to the initial seed.
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An example with n = 3
B , x1, x2, x3
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An example with n = 3
B , x1, x2, x3
B1,x′1,x2, x3
B2, x1,x′2,x3
B3, x1, x2,x′3
18
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An example with n = 3
18
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An example with n = 3
18
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An example with n = 3
18
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An example with n = 3
18
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An example with n = 3
18
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An example with n = 3
Theorem (Fomin, Zelevinsky). Cluster algebras with a finitenumber of seeds correspond to generalized associahedra.
18
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Second take-home lesson
Cluster algebras (of “finite type”) are intricately connected withthe combinatorics of Coxeter groups. But in most contexts,restricting to cluster algebras of finite type is very artificial.
The standard construction of generalized associahedra does notgeneralize in an obvious way to an infinite Coxeter group. It alsouses very little Coxeter theoretic machinery.
For the rest of the talk, I’ll describe an alternate approach togeneralized associahedra that uses a wide range ofCoxeter-theoretic tools and which generalizes naturally to infiniteCoxeter groups.
19
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A map from permutations to triangulations
Arrange the numbers from 0 to n + 1 on a polygon such thatnumbers strictly increase left to right. Begin with a path along thebottom. Modify the path by removing/adding vertices in the ordergiven by π. The triangulation is the union of the paths.
Example: π = 42783165
1
2
3 4
5
6 7
8
90
20
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A map from permutations to triangulations
Arrange the numbers from 0 to n + 1 on a polygon such thatnumbers strictly increase left to right. Begin with a path along thebottom. Modify the path by removing/adding vertices in the ordergiven by π. The triangulation is the union of the paths.
Example: π = 42783165
4
1
2
3 4
5
6 7
8
90
20
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A map from permutations to triangulations
Arrange the numbers from 0 to n + 1 on a polygon such thatnumbers strictly increase left to right. Begin with a path along thebottom. Modify the path by removing/adding vertices in the ordergiven by π. The triangulation is the union of the paths.
Example: π = 42783165
42
1
2
3 4
5
6 7
8
90
20
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A map from permutations to triangulations
Arrange the numbers from 0 to n + 1 on a polygon such thatnumbers strictly increase left to right. Begin with a path along thebottom. Modify the path by removing/adding vertices in the ordergiven by π. The triangulation is the union of the paths.
Example: π = 42783165
427
1
2
3 4
5
6 7
8
90
20
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A map from permutations to triangulations
Arrange the numbers from 0 to n + 1 on a polygon such thatnumbers strictly increase left to right. Begin with a path along thebottom. Modify the path by removing/adding vertices in the ordergiven by π. The triangulation is the union of the paths.
Example: π = 42783165
4278
1
2
3 4
5
6 7
8
90
20
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A map from permutations to triangulations
Arrange the numbers from 0 to n + 1 on a polygon such thatnumbers strictly increase left to right. Begin with a path along thebottom. Modify the path by removing/adding vertices in the ordergiven by π. The triangulation is the union of the paths.
Example: π = 42783165
42783
1
2
3 4
5
6 7
8
90
20
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A map from permutations to triangulations
Arrange the numbers from 0 to n + 1 on a polygon such thatnumbers strictly increase left to right. Begin with a path along thebottom. Modify the path by removing/adding vertices in the ordergiven by π. The triangulation is the union of the paths.
Example: π = 42783165
427831
1
2
3 4
5
6 7
8
90
20
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A map from permutations to triangulations
Arrange the numbers from 0 to n + 1 on a polygon such thatnumbers strictly increase left to right. Begin with a path along thebottom. Modify the path by removing/adding vertices in the ordergiven by π. The triangulation is the union of the paths.
Example: π = 42783165
4278316
1
2
3 4
5
6 7
8
90
20
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A map from permutations to triangulations
Arrange the numbers from 0 to n + 1 on a polygon such thatnumbers strictly increase left to right. Begin with a path along thebottom. Modify the path by removing/adding vertices in the ordergiven by π. The triangulation is the union of the paths.
Example: π = 42783165
42783165
1
2
3 4
5
6 7
8
90
20
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A map from permutations to triangulations
Arrange the numbers from 0 to n + 1 on a polygon such thatnumbers strictly increase left to right. Begin with a path along thebottom. Modify the path by removing/adding vertices in the ordergiven by π. The triangulation is the union of the paths.
Example: π = 42783165
42783165
1
2
3 4
5
6 7
8
90
20
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S4 to triangulations
1234
2134 1324 1243
2314 3124 1342 2143 1423
2341 3214 3142 2413 1432 4123
3241 2431 3412 4213 4132
3421 4231 4312
4321
21
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S4 to triangulations
1234
2134 1324 1243
2314 3124 1342 2143 1423
2341 3214 3142 2413 1432 4123
3241 2431 3412 4213 4132
3421 4231 4312
43210
12 3
4
5
21
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S4 to triangulations
0
12 3
4
5
21
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S4 to triangulations
0
12 3
4
5
21
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S4 to triangulations, geometric view
22
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S4 to triangulations, geometric view
22
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S4 to triangulations, geometric view
22
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S4 to triangulations, geometric view
22
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S4 to triangulations, geometric view
22
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S4 to triangulations, geometric view
22
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Summary of the Sn example
Sn
Image(πc↓)
πc↓
?
clc
- {Triangulations}
ηc
-
c : the choice of how to label the polygon.
ηc : Permutations to triangulations. Factors through πc↓ .
πc↓ : x 7−→ bottom element of η−1
c (ηc(x)).
clc : a bijection from bottom elements to triangulations.
Key point: Combinatorics of associahedron encoded in fibers of πc↓ .
23
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Summary of the Sn example
Sn
Image(πc↓)
πc↓
?
clc
- {Triangulations}
ηc
-
c : the choice of how to label the polygon.
ηc : Permutations to triangulations. Factors through πc↓ .
πc↓ : x 7−→ bottom element of η−1
c (ηc(x)).
clc : a bijection from bottom elements to triangulations.
Key point: Combinatorics of associahedron encoded in fibers of πc↓ .
23
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Third take-home lesson (for general finite W )
W
{c-sortable elements}
πc↓
?
clc
- {Clusters}
ηc (?)
-
c : a choice of Coxeter element of W .
c-sortables and πc↓ defined using weak order and/or reduced words
and/or geometry.
clc : still a bijection.
24
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Third take-home lesson (for general finite W )
W
{c-sortable elements}
πc↓
?
clc
- {Clusters}
ηc (?)
-
c : a choice of Coxeter element of W .
c-sortables and πc↓ defined using weak order and/or reduced words
and/or geometry.
clc : still a bijection. (Also, ncc , a bijection to NC partitions.)
24
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Third take-home lesson (for general finite W )
W
{c-sortable elements}
πc↓
?
clc
- {Clusters}
ηc (?)
-
c : a choice of Coxeter element of W .
c-sortables and πc↓ defined using weak order and/or reduced words
and/or geometry.
clc : still a bijection. (Also, ncc , a bijection to NC partitions.)
Key point: Fibers of πc↓ define Cambrian fan, combinatorially
isomorphic to cluster complex (coincides with “g -vector fan”). 24
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Infinite Coxeter groups/cluster algebras of infinite type
W
{c-sortable elements}
πc↓
?
clc
- {Clusters}
ηc
-
c-sortables and πc↓ : definitions (weak order, reduced words,
geometry) hold verbatim.
clc : No suitable general definition of combinatorial clusters.ncc : No suitable general definition of noncrossing partitions.
Cambrian fan: Conjecturally, a subfan of the g -vector fan.
The issue: Cambrian fan doesn’t reach outside the Tits cone.25
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An “affine” example (W of type G2)
26
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An “affine” example (W of type G2)
26
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An “affine” example (W of type G2)
26
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An “affine” example (W of type G2)
26
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The same affine example (Negative of Tits cone)
27
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The same affine example (stereographic view)
28
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The same affine example (stereographic view)
28
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The same affine example (stereographic view)
28
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The same affine example (stereographic view)
28
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The same affine example (stereographic view)
28
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The same affine example (stereographic view)
28
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Analysis of the affine example
The Cambrian fan becomes periodic in one direction.
Cluster algebras tell us that the dual “affine associahedron” shouldbe combinatorially periodic in both directions.
Part of the associahedron is outside of the Cambrian fan, becausethe Cambrian fan doesn’t extend (very far) outside the Tits cone.
Speyer and I think we know how to deal with the affine case.A key point: all of the combinatorics is already present in the(limiting) periodic part.
The non-affine cases are harder. . .
29
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A hyperbolic example
30
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The same hyperbolic example (stereographic view)
31
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The same hyperbolic example (stereographic view)
31
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The same hyperbolic example (stereographic view)
31
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The same hyperbolic example (stereographic view)
31
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The same hyperbolic example (stereographic view)
31
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The same hyperbolic example (stereographic view)
31
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The same hyperbolic example (stereographic view)
31
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The same hyperbolic example (stereographic view)
31
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Analysis of the hyperbolic example
Speyer and I don’t think we can deal with the non-affine infinitecase with only sortable elements/Cambrian lattices as tools.
There are indications that some larger object (properly containingthe Coxeter group W ) exists. The key may be to generalizesortable elements and Cambrian lattices to this bigger setting.
32
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The geometry of sortable elements
33
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The geometry of sortable elements (continued)
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The word-combinatorics of sortable elements
A Coxeter element is c = s1 · · · sn for S = {s1, . . . , sn}
Fix (some reduced word for) a Coxeter element c .
Form an infinite word
c∞ = c |c |c |c | · · ·
c-Sorting word for w is the the lexicographically leftmost subwordof c∞ which is a reduced word for w .
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 42351
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 42351
36
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 41352
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 41352
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 41253
36
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 41253
36
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 412533 s1s2s3 31254
36
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 412533 s1s2s3 31254
36
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 412533 s1s2s3 312544 s1s2s3s4| 31245
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 412533 s1s2s3 312544 s1s2s3s4| 31245
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 412533 s1s2s3 312544 s1s2s3s4| 312455 s1s2s3s4| 31245
36
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 412533 s1s2s3 312544 s1s2s3s4| 312455 s1s2s3s4| 31245
36
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 412533 s1s2s3 312544 s1s2s3s4| 312455 s1s2s3s4| 312456 s1s2s3s4|s2 21345
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 412533 s1s2s3 312544 s1s2s3s4| 312455 s1s2s3s4| 312456 s1s2s3s4|s2 21345
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 412533 s1s2s3 312544 s1s2s3s4| 312455 s1s2s3s4| 312456 s1s2s3s4|s2 213457 s1s2s3s4|s2 21345
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 412533 s1s2s3 312544 s1s2s3s4| 312455 s1s2s3s4| 312456 s1s2s3s4|s2 213457 s1s2s3s4|s2 21345
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 412533 s1s2s3 312544 s1s2s3s4| 312455 s1s2s3s4| 312456 s1s2s3s4|s2 213457 s1s2s3s4|s2 213458 s1s2s3s4|s2 21345
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 412533 s1s2s3 312544 s1s2s3s4| 312455 s1s2s3s4| 312456 s1s2s3s4|s2 213457 s1s2s3s4|s2 213458 s1s2s3s4|s2 21345
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 412533 s1s2s3 312544 s1s2s3s4| 312455 s1s2s3s4| 312456 s1s2s3s4|s2 213457 s1s2s3s4|s2 213458 s1s2s3s4|s2 213459 s1s2s3s4|s2|s1 12345
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Example: W = S5, c = s1s2s3s4
c∞ = s1s2s3s4|s1s2s3s4|s1s2s3s4| · · ·
The c-sorting word for 42351 is s1s2s3s4|s2|s1.
Step c-Sorting word Permutation
0 423511 s1 413522 s1s2 412533 s1s2s3 312544 s1s2s3s4| 312455 s1s2s3s4| 312456 s1s2s3s4|s2 213457 s1s2s3s4|s2 213458 s1s2s3s4|s2 213459 s1s2s3s4|s2|s1 12345
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Sortable elements
A sorting word can be interpreted as a sequence of sets (lettersbetween dividers “ | ”). If the sequence is nested then w isc-sortable.
Example: w with c-sorting word s1s2s3s4|s2|s1 is not c-sortablebecause {s1} 6⊆ {s2}.
Example: W = S3, c = s1s2.
c-sortable: not c-sortable:1 123 s2|s1 312s1 213s1s2 231s1s2|s1 321s2 132
Example: W = Sn
For one choice of c , the c-sortable elements are the “231-avoiding”or “stack-sortable” permutations.For another c , “c-sortable” = “312-avoiding”.
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