Polyhedral Combinatorics of Coxeter Groups 2
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1 3 s 2 s 6 s 4 s 5 s 1 s 3 s 2 s 6 s 4 s 5 s 1 s 3 s Polyhedral Combinatorics of Coxeter Groups 2 s 6 s 5 s 1 s Dissertation s Defense 3 s 2 s 6 Jean-Philippe Labbé s 1 s s 2 s 1 α u ut(α s ) u(α t ) ut(α u ) July 8 th 2013
2 A few motivations Open Problem (Dyer (1993)) Is there, for each infinite Coxeter group, a complete ortholattice that contains the weak order? Open Problem (Jonsson (2003)) Is there a polytopal realization of the multi-associahedron?
3 A few motivations Open Problem (Dyer (1993)) Is there, for each infinite Coxeter group, a complete ortholattice that contains the weak order? Introduction of the limit roots of an infinite Coxeter group. Open Problem (Jonsson (2003)) Is there a polytopal realization of the multi-associahedron? Introduction of the multi-cluster complex of a finite Coxeter group.
4 PART I: Ortholattice for infinite Coxeter groups? Fixed-gear drivetrain
5 PART I: Ortholattice for infinite Coxeter groups? Fixed-gear drivetrain
6 PART I: Ortholattice for infinite Coxeter groups? Fixed-gear drivetrain Geometry Algebra
7 PART I: Ortholattice for infinite Coxeter groups? Fixed-gear drivetrain Algebra Combinatorics Geometry
8 The weak order on Coxeter groups (W, S) infinite Coxeter group ( s, t S e = s 2 = (st) ms,t ) Definition (Weak order) Let u, v W. Then u v u is a prefix of v. Example tus and tustut.
9 The weak order on Coxeter groups (W, S) infinite Coxeter group ( s, t S e = s 2 = (st) ms,t ) Definition (Weak order) Let u, v W. Then u v u is a prefix of v. Example tus and tustut. (W, ) is a meet-semilattice. There is no maximal element. Question How to tell when the join of two elements exists?
10 Asymptotical behaviour of roots Study the directions of the roots. Q ρ 4 ρ 3 ρ 4 ρ 3 ρ ρ 2 2 β = ρ 1 ρ ρ 2 2 α = ρ 1 ρ 1 ρ 1 V 1 V 1 Q The infinite dihedral group I 2 ( ).
11 Limit roots of Coxeter groups s α s δ s β s α s β s γ s δ s γ
12 Limit roots of Coxeter groups s α s δ s β s α s β s γ s δ s γ Theorem (Hohlweg-L.-Ripoll, Canad. J. Math. (2013)) The set E(Φ) of accumulation points of normalized roots Φ is contained in the isotropic cone of (V, B).
13 A first step using biconvexity Theorem (L. 2012) Let (W, S) be a Coxeter group of rank n 3, there is a complete ortholattice containing the weak order. The join of two elements can be computed using convex hulls.
14 A first step using biconvexity Theorem (L. 2012) Let (W, S) be a Coxeter group of rank n 3, there is a complete ortholattice containing the weak order. The join of two elements can be computed using convex hulls. Fact (L. 2012) For Coxeter groups of rank n 4 the join can not be computed using convex hulls. The notion of convexity is to restrictive. biclosedness seems to be the right geometry to look at.
15 PART II: Subword Complexes in Discrete Geometry Algebra Combinatorics Geometry
16 Multi-cluster complexes triangulations
17 Multi-cluster complexes triangulations Finite Coxeter Groups (Algebraic) Integer k > 1 (Geometric)
18 Multi-cluster complexes Finite Coxeter Groups (Algebraic) triangulations gen. triangulations clusters Integer k > 1 (Geometric) multi-triangulations
19 Multi-cluster complexes Finite Coxeter Groups (Algebraic) triangulations gen. triangulations clusters Integer k > 1 (Geometric) multi-triangulations gen. multi-triangulations multi-clusters
20 Simplicial complex of multi-triangulations Fix a convex m-gon. Multi-triangulation: Maximal set of diagonals not containing k + 1 pairwise crossing diagonals.
21 Simplicial complex of multi-triangulations Fix a convex m-gon. Multi-triangulation: Maximal set of diagonals not containing k + 1 pairwise crossing diagonals. m,k : the simplicial complex with faces sets of (relevant) diagonals not containing k + 1 pairwise crossing diagonals
22 Simplicial complex of multi-triangulations Fix a convex m-gon. Multi-triangulation: Maximal set of diagonals not containing k + 1 pairwise crossing diagonals. m,k : the simplicial complex with faces sets of (relevant) diagonals not containing k + 1 pairwise crossing diagonals Conjecture (Jonsson, 2003) m,k is isomorphic to the boundary complex of a simplicial polytope.
23 Simplicial complex m,k - Example Let m = 6 and k = 2 When m = 2k + 2, m,k is a k-simplex.
24 Simplicial complex m,k - Example Let m = 6 and k = 2 When m = 2k + 2, m,k is a k-simplex.
25 Subword complexes (W, S) finite Coxeter system of rank n Definition (Knutson-Miller, 2004) Let Q = (q 1,..., q r ) be a word in S and π W. simplicial complex subword complex (Q, π) := facets complements (in Q) of reduced expressions of π
26 Subword complexes (W, S) finite Coxeter system of rank n Definition (Knutson-Miller, 2004) Let Q = (q 1,..., q r ) be a word in S and π W. simplicial complex subword complex (Q, π) := facets complements (in Q) of reduced expressions of π Theorem (Knutson-Miller, 2004) Subword complexes are topological spheres or balls.
27 Multi-cluster complex - Definition Definition (Ceballos-L.-Stump, J. of Alg. Comb. 2013) The multi-cluster complex k c (W ) is the subword complex (c k w (c), w ) of type W.
28 Multi-cluster complex - Definition Definition (Ceballos-L.-Stump, J. of Alg. Comb. 2013) The multi-cluster complex k c (W ) is the subword complex (c k w (c), w ) of type W. Theorem (CLS, 2013) The subword complex (cw (c), w ) is isomorphic to the c-cluster complex of type W.
29 Multi-cluster complexes of type A and B Theorem (Pilaud-Pocchiola 2012, Stump 2011) The multi-cluster simplicial complex complex k c (A n ) = of k-triangulations of a convex m-gon where m = n + 2k + 1.
30 Multi-cluster complexes of type A and B Theorem (Pilaud-Pocchiola 2012, Stump 2011) The multi-cluster simplicial complex complex k c (A n ) = of k-triangulations of a convex m-gon where m = n + 2k + 1. Theorem (CLS, 2013) The multi-cluster simplicial complex of centrally complex k c (B m k ) = symmetric k-triangulations of a regular convex 2m-gon Corollary is a vertex-decomposable simplicial sphere. sym m,k
31 Universality and polytopality of k c(w ) Question (Knutson-Miller, 2004) Charaterize all simplicial spheres that can be realized as a subword complex.
32 Universality and polytopality of k c(w ) Question (Knutson-Miller, 2004) Charaterize all simplicial spheres that can be realized as a subword complex. Theorem (CLS, 2013) A simplicial sphere it is the link is realized as of a face of a multi-cluster a subword complex complex k c (W ).
33 Universality and polytopality of k c(w ) Question (Knutson-Miller, 2004) Charaterize all simplicial spheres that can be realized as a subword complex. Theorem (CLS, 2013) A simplicial sphere it is the link is realized as of a face of a multi-cluster a subword complex complex k c (W ). Corollary The following two statements are equivalent. (i) Every spherical subword complex is polytopal. (ii) Every multi-cluster complex is polytopal.
34 Polytopality of multi-cluster complexes Conjecture ( Knutson-Miller 04, Jonsson 05, Soll-Welker 09) The multi-cluster complex is the boundary complex of a simplicial polytope.
35 Polytopality of multi-cluster complexes Conjecture ( Knutson-Miller 04, Jonsson 05, Soll-Welker 09) The multi-cluster complex is the boundary complex of a simplicial polytope. True for k = 1: Chapoton-Fomin-Zelevinsky, Hohlweg-Lange-Thomas, Pilaud-Stump, Stella True for I 2 (m), k 1: cyclic polytope, Ceballos-L.-Stump True for A 3, k = 2: Bokowski-Pilaud, Ceballos-L.
36 Polytopality of multi-cluster complexes Conjecture ( Knutson-Miller 04, Jonsson 05, Soll-Welker 09) The multi-cluster complex is the boundary complex of a simplicial polytope. True for k = 1: Chapoton-Fomin-Zelevinsky, Hohlweg-Lange-Thomas, Pilaud-Stump, Stella True for I 2 (m), k 1: cyclic polytope, Ceballos-L.-Stump True for A 3, k = 2: Bokowski-Pilaud, Ceballos-L. Recent work of Bergeron-Ceballos-L.: New combinatorial construction giving realizations of A 3, k = 2 Fan realizations of all subword complexes of type A 3
37 Thanks! s1s2s3s1s2s3s1s2s3s1s2s3s1s2s3 Merci! Thank you! Grazie! Danke! Gracias! s1s2s3s1s2s3s1s2s3s1s2s3s1s2s3 e
38 Recent developments on limit roots Relation with the Tits cone [Dyer-Hohlweg-Ripoll, (arxiv:2013)] Limit set of Kleinian groups [Hohlweg-Préaux-Ripoll, (arxiv:2013)] Sphere packings and geometric invariants for Coxeter groups [Chen-L., (in preparation)]
39 s1s2s3s1s2s3s1s2s3s1s2s3s1s2s3 Recent developments on multi-cluster complexes s1s2s3s1s2s3s1s2s3s1s2s3s1s2s3 e Generalized brick polytope, spanning trees [Pilaud-Stump, arxiv: ] Denominator vectors of cluster algebras of finite types [Ceballos-Pilaud, (arxiv:2013)] Common vertices of permutahedra and generalized associahedra [L., 2013 and L.-Lange (in preparation)]
40 Geometric computation of the join u 5 5 α u s 5 t ŝ(α u) û(α s) û(α t) ût(α s) ût(α u) su(α t) α s α t
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