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On generating functions and automata associated to reflections in Coxeter systems

Riccardo Biagioli, Christophe Hohlweg, Elisa Sasso

Abstract

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.

On generating functions and automata associated to reflections in Coxeter systems

Abstract

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 -canonical automata associated with -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.
Paper Structure (22 sections, 34 theorems, 88 equations, 7 figures)

This paper contains 22 sections, 34 theorems, 88 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Combinatorics of words and roots in Coxeter systems
  3. Combinatorics of words in Coxeter systems
  4. Garside shadows automata
  5. Root systems and inversion sets
  6. The root poset
  7. Small roots
  8. The dominance order
  9. $m$-small roots and $m$-low elements
  10. Reflection-prefixes and palindromic reduced words of reflections
  11. Reflection-prefixes
  12. Reflection-prefixes and the root poset
  13. Reflection-prefixes and the dominance order
  14. Reflection-prefixes and dihedral reflection subgroups
  15. Reflection-prefixes and canonical generators of dihedral reflection subgroups
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $(W,S)$ be a Coxeter system.

Figures (7)

  • Figure 1: The automaton $\mathcal{A}_{G}$ of Example \ref{['ex:aut1']}.
  • Figure 2: Hasse diagram of the root poset of Example \ref{['ex:root poset']}.
  • Figure 3: The saturated chains corresponding to the $t$-prefixes of Example \ref{['ex:a3-chains']}.
  • Figure 4: The automaton $\mathcal{A}_0(W,S)$ of Example \ref{['ex:auta2']} for type $\widetilde{A}_2$ which recognizes the language of reflection-prefixes of $W$.
  • Figure 5: Hasse diagram of the root poset of type $\widetilde{A}_2$ up to depth 3, with $\Sigma=\Sigma_0$ and $\Sigma_1$ that correspond to the blocks connected with non-dotted edges.
  • ...and 2 more figures

Theorems & Definitions (82)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Theorem 2.2: HNW
  • Example 2.3
  • Lemma 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7
  • ...and 72 more