The Generating Function for the Number of Roots of a Coxeter Group

Abstract

Using elementary roots and finite automata, we show that the generating function counting the depths of the roots of a Coxeter group of finite rank is rational. c 1999 Academic Press Root SystemsLet (W, R) be a Coxeter system of finite rank |R|. Then W = r ∈ R : (rs) mrs = 1 for r, s ∈ R , where m rr = 1 and m rs ≥ 2 for r, s ∈ R, r = s (with m rs = ∞ allowed). Let Π = {α r : r ∈ R} be the basis of an R-vector space V . We define a bilinear symmetric form on V by (α r , α s ) = − cos(π/m rs ) (and (α r , αThe set Φ = {w · α r : w ∈ W, r ∈ R} ⊂ V is called the root system of W . Let Φ + = { c r α r ∈ Φ : c r ≥ 0 for all r ∈ R} denote the set of positive roots, and putBy (w), the length of w, we denote the minimal integer n for which there exists an expression w = r 1 · · · r n with r i ∈ R. An expression with n = (w) is called reduced.The following facts are well known and can be found in Humphreys (1990). Proposition 1.1. (i) For w ∈ W and r ∈ R we haveFor α ∈ Φ + the depth dp(α) of α is defined as dp(α) = min{ (w) :Using r(Φ + \ {α r }) = Φ + \ {α r }, it is easily seen that dp(α) equals the minimum value of (w) + 1 taken over all pairs (w, r) ∈ W × R satisfying w · α r = α.For α ∈ Φ, the reflection r α with respect to α is defined as r α · v = v − 2(v, α)α. For w ∈ W , r ∈ R with w · α r = α, we have r α = wrw −1 by the W -invariance of (·, ·). Hence r α ∈ W . The following lemma is Corollary 2.7 of Brink (1994).

On the length of fully commutative elements

“…This is classically shown by induction using elementary combinatorics of Coxeter groups, see [4,Section 7.1] or [16,Section 5.12]. A lesser known result is that this is true for the set of reflections of W , which are the conjugates of generators S [7]. Another (simpler) example is that elements possessing a unique reduced expression also have a rational generating function [6].…”

“…1 − q 2 − q 3 − 2q 6 − q 7 + 3q 8 + q 9 + q 12 0.9225155924 Y 6 1 − q 12 − q 17 − q 25 0.9378483025 Y 7 1 − 3q 3 − q 7 0.6778224161 Y 8 1 − q 12 − q 17 − q 25 0.9378483025 Y 9 1 − q 18 − q 37 0.9740122556 Figure 7. For each group W of type Y i illustrated in Figure 6, we compute the denominator of the rational function…”

Minimal inversion complete sets and maximal abelian ideals

“…. , α 4 }, with numbering as in [13], then w = σ 3 σ 2 σ 3 σ 4 is the shortest element taking θ = 2α 1 + 4α 2 + 3α 3 + 2α 4 to µ = 2α 1 + 2α 2 + α 3 + α 4 .…”

“…This notion has been studied in the context of arbitrary (possibly infinite) Coxeter groups [2,4], but we only use it for Weyl groups. It is easily seen that if the required minimum of ℓ(w) is achieved, then w(γ) ∈ −Π.…”

Involution products in Coxeter groups

“…In [6] expressions for L W (Bn) (t) and L W (Dn) (t) are given, but unfortunately the proofs contain errors which lead to the results being incorrect. Finally we remark that the length polynomial for the special case where X is the set of reflections in a Coxeter group W has been studied in another guise [4]. Theorem 4.1 of [4] gives the generating function for counting the depth of roots of a Coxeter system of finite rank.…”