In this article, we study two combinatorial problems concerning the set of reflections of a Coxeter system. The first problem asks whether the language of palindromic reduced words for reflections is regular, and the second is about finding formulas for the Poincaré series of reflections, namely the generating function of reflection lengths. These two problems were inspired by a conjecture of Stembridge stating that the Poincaré series of reflections is rational and by the solution provided by de Man. To address the first problem, we introduce reflection-prefixes, arising naturally from palindromic reduced words. We study their connections with the root poset, dominance order on roots, and dihedral reflection subgroups. Using $m$-canonical automata associated with $m$-Shi arrangements, we prove that the language of reduced words for reflection-prefixes is regular. For the second problem, we focus on affine Coxeter groups. In this case, we derive a simple formula for the Poincaré series using symmetries of the Hasse diagram of the root poset.
Let (W, S) be a Coxeter system, with S finite. We denote by S * the free monoid on the alphabet S. To distinguish between a word and its corresponding group element, we denote a word in S * using bold letters, w = s 1 • • • s k , while the resulting product in W is denoted by w = s 1 • • • s k . Let w ∈ W , a word w = s 1 • • • s k ∈ S * is a reduced word for w if w = s 1 • • • s k and k is minimal for this property; in this case the length of w is ℓ(w) = k. The identity e ∈ W is represented by the empty word, and ℓ(e) = 0. We denote the set of all reduced words of w ∈ W by Red(w), and the set of all reduced words in W by Red = w∈W Red(w), where ⊔ denotes the disjoint union. Finally, for any subset A ⊆ W , the Poincaré series of A is the formal power series A(q) := w∈A q ℓ(w) .
In this article, we study two combinatorial problems regarding the set of reflections T := w∈W wSw -1 of a Coxeter group W :
(1) Is the language of palindromic reduced words (for the reflections) regular? (2) Are there explicit and elegant formulas for the Poincaré series of the set T of reflections ?
Our main contributions to those questions are the following:
• We introduce the notion of reflection-prefixes, a class of elements in W arising naturally from palindromic reduced words of reflections, and study their properties in relation to the root poset, the dominance order on roots and dihedral reflection subgroups. • For any Coxeter system, we show that the language of reduced words for reflection-prefixes, Pref T , is regular. This is achieved using the family of m-canonical automata associated with m-Shi arrangements (see [15, §3.4] and [6]), which provide a family of finite deterministic automata recognizing Pref T . As a consequence, we show that the generating function of palindromic reduced words is rational. • In the case of affine Coxeter groups, we derive a simple expression for the Poincaré series of the set of reflections in terms of rational fractions. This formula is obtained from some symmetries of the Hasse diagram of the root poset. We discuss now some history and motivations that lead to this article. If W is finite, A(q) is clearly a polynomial for any A ⊆ W . A natural question in combinatorics of infinite Coxeter systems is to classify for which subsets A the Poincaré series A(q) is rational, that is, can be written as a ratio of two polynomials in q. It is well-known that W (q) is rational, see for instance [1,Corollary 7.1.8]. Furthermore, an explicit recursive formula for W (q) in term of standard parabolic subgroups is provided in [1,Proposition 7.1.7].
A further natural direction is the study of generating functions of words in S * in relation to the Coxeter system (W, S). More precisely, given the canonical projection π : S * → W sending a word w = s 1 • • • s k to the element w = s 1 • • • s k , it is natural to consider the “lifted” Poincaré series of a subset B ⊆ S * relative to (W, S): B(q) = w∈B q ℓ(w) ,
where the notation ℓ(w) is understood to be ℓ
A longstanding open problem in combinatorics of Coxeter groups is the enumeration of |Red(w)|; for a detailed discussion see [1, p.123], and for various partial results, see [12,14]. However, if we consider the set of all elements of a given length, we know that the numbers r k of reduced words of length k ∈ N is enumerated by the following Poincaré series Red(q) = w∈Red q ℓ(w) = k∈N r k q k , which is known to be rational. The proof follows from the existence of a finite deterministic automaton that recognizes the language Red; see for instance [1, Theorem 4.9.1] for more details.
As reported by Brenti [3], Stembridge proposed the following problem during an open problem session at the Mathematical Sciences Research Institute at Berkeley in 1997: Is it true that the Poincaré series T (q) = t∈T q ℓ(t) is rational? Since reflections are known to admit palindromic reduced words, a natural approach to this question is to construct a finite automaton that recognizes exactly one palindromic reduced word for each reflection. The existence of such a regular language would imply that the associated generating function is rational. However, the property of being regular is not generally preserved under the palindromic constraint. If a language is regular, it is well-known that the sublanguage of its palindromic words is context-free but not necessarily regular; for instance if S = {a, b} then S * is regular but a standard application of the pumping lemma shows that the sublanguage of its palindroms is not. Indeed, the regularity of Pal, the language of all palindromic reduced words, which are necessarily reduced words for reflections, remains an open question. In a recent article, Milićević [17] provides palindromic reduced words for all reflections in finite Weyl groups, but highlights the challenge of finding a general algorithm applicable to any Coxeter group. To overcome these challenges, we shift to the language of reflection-prefixes which we define
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