The Distance Spectra of of Coxeter Groups Paul Renteln 1 Department of Physics California State University San Bernardino 2 Department of Mathematics California Institute of Technology ECNU, Shanghai June 7, 2010 Outline Open Problems Root Systems Finite Reflection Group V an n dimensional inner product space over R Given α V, reflection t α : V V fixes hyperplane H α := {v V : (α, v) = 0} (pointwise) and sends α to α. Let Φ V satisfy tα Φ = Φ for all α Φ Φ Rα = { α, α} for all α Φ Φ is called a root system. Let Φ be an irreducible finite root system Let W be the group generated by reflections t α, α Φ W called a finite reflection group or finite Coxeter group Four infinite families: A n, B n, D n, I 2 (m), and six exceptional types: E 6, E 7, E 8, F 4, H 3, H 4
Reflection Groups Arising in Nature Simple Systems S 4 = W (A 3 ) W (B 3 ) W (H 3 ) Φ is a set of simple roots if: is a basis for V Every root α Φ can be written α = α i c iα i where all c i 0 or all c i 0 The simple reflections are S := {s α : α } Source: http://en.wikipedia.org/wiki/regular polytope Ex: S 3 = W (A 2 ) Symmetries of a triangle Length and Absolute Length hyperplanes simple roots roots Let T := {set of all reflections} Let S := {set of simple reflections} Every element w W is a word in the reflections in T or S The minimum length l T (w) of w W written as a word in T is called the absolute length of w. The minimum length l S (w) of w W written as a word in S is called the length of w. Note: l T (w) is constant on conjugacy classes, while l S (w) is not
Coxeter Transformations The Coxeter element of W is c = s S s (unique up to conjugacy) The order of c is h, the Coxeter number The eigenvalues of c (in the reflection representation) are of the form ζ m i for some primitive h th root of unity ζ. The numbers m 1, m 2,... m n are the exponents of W The Exponents of the Reflection Groups Type m 1, m 2,..., m n A n 1, 2,..., n B n 1, 3, 5,..., 2n 1 D n 1, 3, 5,..., 2n 1, n 1 E 6 1, 4, 5, 7, 8, 11 E 7 1, 5, 7, 9, 11, 13, 17 E 8 1, 7, 11, 13, 17, 19, 23, 29 F 4 1, 5, 7, 11 G 2 1, 5 H 3 1, 5, 9 H 4 1, 11, 19, 29 I 2 (m) 1, m 1 Poincaré Polynomials Theorem (Chevalley 55, Solomon 66, Steinberg 68) w W q l S(w) = n i=1 1 q m i+1 1 q Theorem (Shephard & Todd 54, Solomon 63) w W q l T (w) = n (1 + m i q) i=1 W a finite group R W a symmetric subset of generators without identity: Γ(W, R) a Cayley graph: {r W : r R r 1 R, r 1} V (Γ) = W E(Γ) = {{w, wr} : r R} A Cayley graph Γ(W, R) is normal if R is closed under conjugation
Adjacency Spectrum Distance Spectrum Let u, v V (Γ). The adjacency matrix: A(u, v) = 1 if {u, v} E(Γ) and 0 otherwise. The adjacency spectrum is the set of eigenvalues of A Some known adjacency spectra: Γ(S n, all transpositions) (= Γ(S n, T )) Diaconis & Shahshahani, 81 Γ(S n, star transpositions) Flatto, Odlyzko, and Wales, 85 Γ(S n, adjacent transpositions) (= Γ(S n, S)) Bacher, 94 (part) Γ(S n, derangements), 07 Let u, v V (Γ). The distance d(u, v) = the length of a shortest path from u to v The distance polynomial is the characteristic polynomial of d considered as a matrix The distance spectrum is the set of eigenvalues of d, i.e., roots of the distance polynomial Absolute Order Graph Absolute Order Graph on W : Γ(W, T ) where T = {all reflections}. Normal Example: (123) (12) (13) 1 (132) (23) 1 (12) (123) (13) (132) (23) 1 0 1 2 1 2 1 (12) 1 0 1 2 1 2 (123) 2 1 0 1 2 1 (13) 1 2 1 0 1 2 (132) 2 1 2 1 0 1 (23) 1 2 1 2 1 0 Weak Order Graph (123) (12) Weak Order Graph on W : Γ(W, S) where S = {simple reflections}. Not normal Example: (13) 1 (132) (23) 1 (12) (123) (13) (132) (23) 1 0 1 2 3 2 1 (12) 1 0 1 2 3 2 (123) 2 1 0 1 2 3 (13) 3 2 1 0 1 2 (132) 2 3 2 1 0 1 (23) 1 2 3 2 1 0 Absolute Order Graph Γ(S 3, T ) Distance Matrix d Weak Order Graph Γ(S 3, S) Distance Matrix d
Distance Polynomials of Some Absolute Order Graphs Type Distance Polynomial A 2 (z 7)(z 1)(z + 2) 4 A 3 (z 46)(z 2) 9 (z + 2) 5 (z + 6) 9 A 4 (z 326)(z 6) 37 (z 2) 25 (z + 4) 16 (z + 6) 25 (z + 24) 16 B 3 z 9 (z 100)(z 4) 14 (z + 4) 9 (z + 8) 15 H 3 (z 268)(z 8) 18 (z 4) 25 (z +2) 32 (z +8) 25 (z +12) 18 (z +32) D 4 z 27 (z 544)(z 16)(z 8) 36 (z 4) 64 (z + 8) 20 (z + 16) 27 (z + 32) 16 F 4 z 160 (z 3600)(z 240)(z 48) 2 (z 32) 36 (z 24) 16 (z 16) 162 (z 12) 256 (z 8) 144 (z + 16) 117 (z + 24) 128 (z + 48) 105 (z + 96) 24 I 2 (5) (z 13)(z 3)(z + 2) 8 I 2 (6) (z 16)(z 4)(z + 2) 10 I 2 (7) (z 19)(z 5)(z + 2) 12 Generalized Poincaré Polynomials and the Distance Spectrum For each χ Irr(W ) introduce generalized Poincaré polynomial: P χ (q) := 1 χ(1) w W χ(w)q l T (w) Theorem The eigenvalues of the absolute order graphs Γ(W, T ) are given by η χ = dp χ(q) dq, q=1 with multiplicity χ(1) 2 for each χ Irr(W ). Moreover, η χ Z. Proof Sketch Proof Sketch cont. Consider L := w W l T (w)w CW. l T is a conjugacy class invariant, so L T = l T (w K ) w = K w K K l T (w K )I K, Let ρ be left regular representation of W extended to CW. Then L is represented by distance matrix: ρ(l T )v = w = u l T (w)ρ(w)v = w l T (uv 1 )u = u (Last equality follows from bi-invariance of d.) l T (w)wv d(u, v)u. where I K is (up to constant) the trivial idempotent on class K, and hence constant on irreducible modules, by Schur s lemma. The irreducible modules appear with multiplicity equal to their dimensions, so taking traces yields the eigenvalues η χ = 1 χ(1) K K l T (w K )χ(w K ) = dp χ(q) dq, q=1 each with multiplicity χ(1) 2. Integrality follows by a theorem of Isaacs.
Leading (?) Eigenvalue of Absolute Order Graphs P 1 (q) = ordinary Poincaré polynomial = i (1 + m iq) Corollary η 1 = W i m i 1 + m i. Conjecture η 1 is the largest eigenvalue of the distance matrix of the absolute order graphs. Generalized Poincaré Polynomial in Type A n 1 Can obtain remaining eigenvalues without knowing characters explicitly Theorem (Molchanov 82, Stanley EC2) The generalized Poincaré polynomial for the symmetric group S n = W (A n 1 ) is P λ (q) = u λ(1 + qc(u)), where λ n is a partition of n, and c(u) is the content of box u (namely j i if box u is at (i, j) in Ferrers diagram of partition). Multiplicity of η λ is f 2 λ, where f λ = n!/h λ and H λ is the product of all the hook lengths of λ. Generalized Poincaré Polynomials in Other Types Theorem (Molchanov 82) The generalized Poincaré polynomial for the hyperoctachedral group W (B n ) is Γ(S 4, S): The Permutahedron in R 3 P (λ,µ) = u λ [1 + q(c(u) + 1)] v µ[1 + q(c(v) 1)], where λ k, µ m, and n = k + m. Multiplicity of η (λ,µ) is f 2 λµ, where f λµ = n! k!m! f λf µ. Molchanov also computed generalized Poincaré polynomials in types D n and I 2 (n) Source: http://www.antiquark.com/math/permutahedron 4.gif
Distance Polynomials of Some Weak Order Graphs Type Distance Polynomial A 2 z 2 (z 9)(z + 1)(z + 4) 2 A 3 z 17 (z 72)(z + 4) 3 (z + 20) 3 A 4 z 109 (z 600)(z + 20) 6 (z + 120) 4 B 3 z 38 (z 216)(z + 8) 3 (z 2 + 64z + 384) 3 D 4 z 179 (z 1152)(z + 224) 4 (z + 32) 8 D 5 z 1899 (z 19200)(z + 2880) 5 (z + 320) 15 H 3 z 104 (z 900)(z 2 + 248z + 3856) 3 (z + 24) 4 (z + 12) 5 F 4 z 1127 (z 13824)(z + 192) 16 (z 2 + 2688z + 313344) 4 E 6 z 51803 (z 933120)(z + 112320) 6 (z + 8640) 30 I 2 (5) z 4 (z 25)(z + 1)(z 2 + 12z + 16) 2 I 2 (6) z 5 (z 36)(z + 2) 2 (z 2 + 16z + 16) 2 I 2 (7) z 6 (z 49)(z + 1)(z 3 + 24z 2 + 80z + 64) 2 Distance Polynomials of Some Weak Order Graphs Type Distance Polynomial A 2 z 2 (z 9)(z + 1)(z + 4) 2 A 3 z 17 (z 72)(z + 4) 3 (z + 20) 3 A 4 z 109 (z 600)(z + 20) 6 (z + 120) 4 B 3 z 38 (z 216)(z + 8) 3 (z 2 + 64z + 384) 3 D 4 z 179 (z 1152)(z + 224) 4 (z + 32) 8 D 5 z 1899 (z 19200)(z + 2880) 5 (z + 320) 15 H 3 z 104 (z 900)(z 2 + 248z + 3856) 3 (z + 24) 4 (z + 12) 5 F 4 z 1127 (z 13824)(z + 192) 16 (z 2 + 2688z + 313344) 4 E 6 z 51803 (z 933120)(z + 112320) 6 (z + 8640) 30 I 2 (5) z 4 (z 25)(z + 1)(z 2 + 12z + 16) 2 I 2 (6) z 5 (z 36)(z + 2) 2 (z 2 + 16z + 16) 2 I 2 (7) z 6 (z 49)(z + 1)(z 3 + 24z 2 + 80z + 64) 2 Observations A Variant of the Distance Matrix Γ(W, S) not normal, so no simple formula involving characters Lots of zeros (huge degeneracy) and very few distinct eigenvalues! Spectrum not integral in general, but... Conjecture The distance spectra of the weak order graphs of types A, D, and E each contain only four distinct integral eigenvalues. Have partial proof of conjecture and explanations for some of these observations Difficulty is that distance matrix is too large: size W W Instead of working with group elements, can work with roots Use Coxeter combinatorics to find a smaller matrix with size Φ Φ and related spectrum
The Angle Operator Proof Idea Let Ψ = { α : α Φ} be the permutation representation of W Define angle operator Θ by Let Π be set of positive roots (all c i 0). Well known that l S (w) = wπ ( Π), Θ β = α θ αβ α, so d S (u, v) = l S (u 1 v) = vπ u( Π). where θ αβ is the angle between α and β. Theorem The distance spectrum of the weak order graphs can be recovered from the spectrum of Θ. Now define d S (u, v) = uπ vπ. Can show Spec d S = {λ 1, λ 2,..., λ W } Spec d S = {λ 1, λ 2,..., λ W }. Proof Idea cont. Proof Idea cont. Define ψ w := α>0 wα. It follows that d S (u, v) = ψ u ψ v. This has same spectrum as D = ψ w ψ w. w Can show that Show for dihedral groups that D = W (π Θ), 2π then show it holds for all other reflection groups by considering cosets of dihedral subgroups. D αβ = {w W : wα > 0 and wβ > 0}.
Cyclic Eigenvectors in Types A, D, and E A Combinatorial Problem Theorem Let (α, β, γ) Φ be distinct, coplanar roots with α + β + γ = 0. Define cyclic subspace Ψ c Ψ to be space spanned by vectors of form ( ) ( ) ψ αβγ := α + β + γ α + β + γ. Let W be of type A, D, or E. Then Θ ψ αβγ = 2π 3 ψ αβγ. Problem What is the dimension of Ψ c? In type A n 1 roots are of form e i e j for 1 i < j n. Can map roots to directed edges in complete graph K n. Ψ c can be identified with cycle space of K n, which has dimension ( ) n 1 2. In type D can show dim Ψ c = n(n 2). (Proof by representation theory.) Combinatorial argument desired. Open Problems Distance Polynomial of Weak Order Graph in Type A Things To Do Theorem The distance polynomial of Γ(S n, S) is ( q n! (1/2)(n2 n+2) q n! ( )) n 2 2 ( q + n! 6 ) (n 1)(n 2)/2 ( q + ) (n + 1)! n 1. 6 Distance polynomial in type D obtained similarly. (Type I also known.) What about the other types? Applications? Chemistry? Can similar methods help to determine the adjacency spectrum of weak order graphs? Proof. By constructing remaining eigenvectors.