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Bounded weight functions on regular languages and groups

J. Guilhot, E. Little, J. Parkinson

TL;DR

This work develops a theory of bounded weight functions on languages and finitely generated groups, proving that for regular languages the bounded-weight cone is a rational polyhedral cone with a finite witness set, and that the bound-attaining cell is a nonempty regular language with an explicit automaton. It then transfers these ideas to groups via geodesic languages, yielding finite-test characterizations of boundedness when regular geodesic languages exist. Specializing to Coxeter groups, the authors describe the minimal shortlex automata, classify weight functions by parity constraints, and compute explicit bounds and cells for spherical, affine, and triangle groups, with concrete examples. Finally, they connect bounded representations of weighted Hecke algebras to these language-theoretic constructions, showing that bounded 1-dimensional representations yield regular Kazhdan-Lusztig cells and offering insights relevant to Casselman-type conjectures.

Abstract

We introduce the notion of a bounded weight function on a language, and show that the set of bounded weight functions on a regular language is a rational polyhedral cone. We study the cell recognised by a bounded weight function (that is, the set of elements of the language where the bound is attained), and show that if the language is regular then this cell is regular. The related notion of a weight function on a finitely generated group is introduced, and the case of Coxeter groups is studied in detail. Applications to the representation theory of weighted Hecke algebras are given.

Bounded weight functions on regular languages and groups

TL;DR

This work develops a theory of bounded weight functions on languages and finitely generated groups, proving that for regular languages the bounded-weight cone is a rational polyhedral cone with a finite witness set, and that the bound-attaining cell is a nonempty regular language with an explicit automaton. It then transfers these ideas to groups via geodesic languages, yielding finite-test characterizations of boundedness when regular geodesic languages exist. Specializing to Coxeter groups, the authors describe the minimal shortlex automata, classify weight functions by parity constraints, and compute explicit bounds and cells for spherical, affine, and triangle groups, with concrete examples. Finally, they connect bounded representations of weighted Hecke algebras to these language-theoretic constructions, showing that bounded 1-dimensional representations yield regular Kazhdan-Lusztig cells and offering insights relevant to Casselman-type conjectures.

Abstract

We introduce the notion of a bounded weight function on a language, and show that the set of bounded weight functions on a regular language is a rational polyhedral cone. We study the cell recognised by a bounded weight function (that is, the set of elements of the language where the bound is attained), and show that if the language is regular then this cell is regular. The related notion of a weight function on a finitely generated group is introduced, and the case of Coxeter groups is studied in detail. Applications to the representation theory of weighted Hecke algebras are given.
Paper Structure (15 sections, 18 theorems, 40 equations, 7 figures)

This paper contains 15 sections, 18 theorems, 40 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm1:cones']}
  3. Languages and automata theory
  4. Weight functions on a language
  5. Proof of Theorem \ref{['thm1:cones']}
  6. Weight functions on groups
  7. Definitions
  8. Geodesic languages
  9. Coxeter groups
  10. The minimal automata recognising $\mathcal{L}_{\mathsf{lex}}(W,S)$
  11. Weight functions on Coxeter groups
  12. Triangle groups
  13. Spherical Coxeter groups
  14. Affine Coxeter groups
  15. Bounded representations of Hecke algebras

Key Result

Theorem 1

Let $\mathcal{L}$ be a regular language.

Figures (7)

  • Figure 1: Automata recognising $\Gamma_{\mathcal{L}}(\varphi)$
  • Figure 2: The automaton $\mathcal{A}_{\mathsf{lex}}$ for the triangle group $\Delta(3,3,3)$ (see Example \ref{['ex:333b']})
  • Figure 3: The automata $\mathcal{A}_{\mathsf{lex}}$ for the triangle group $\Delta(2,4,6)$
  • Figure 4: An automaton recognising $\Gamma_{W,S}(\varphi_3)$
  • Figure 5: The minimal shortlex automata for $\Delta(2,3,2m)$, $m\geq 3$
  • ...and 2 more figures

Theorems & Definitions (50)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • ...and 40 more