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).
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