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Some infinite-dimensional representations of certain Coxeter groups

Hongsheng Hu

TL;DR

The paper addresses the existence of infinite-dimensional irreducible complex representations for infinite, non-affine Coxeter groups, a phenomenon absent for finite or affine cases. It develops two constructive schemes linked to the topology of the Coxeter graph: (i) when the graph has at least two circuits, it glues dihedral-subgroup representations along the graph’s fundamental group to build a large $W$-module whose irreducible quotient is infinite-dimensional; (ii) when a circuit with an edge label $m_{st}\ge4$ exists, it leverages the universal cover of the graph to produce an analogous representation and extract an irreducible infinite-dimensional quotient. A third, concrete tree-graph example (isomorphic to $\mathrm{PGL}(2,\mathbb{Z})$) demonstrates the method in a different combinatorial setting. An appendix sketches a general argument showing that every infinite non-affine Coxeter group has finite-index subgroups surjecting onto a non-abelian free group, enabling an induced irreducible infinite-dimensional representation. Altogether, the work provides explicit constructions that link graph topology to representation theory and yields new infinite-dimensional irreducible representations for broad classes of Coxeter groups.

Abstract

A Coxeter group admits infinite-dimensional irreducible complex representations if and only if it is not finite or affine. In this paper, we provide a construction of some of those representations for certain Coxeter groups using some topological information of the corresponding Coxeter graphs.

Some infinite-dimensional representations of certain Coxeter groups

TL;DR

The paper addresses the existence of infinite-dimensional irreducible complex representations for infinite, non-affine Coxeter groups, a phenomenon absent for finite or affine cases. It develops two constructive schemes linked to the topology of the Coxeter graph: (i) when the graph has at least two circuits, it glues dihedral-subgroup representations along the graph’s fundamental group to build a large -module whose irreducible quotient is infinite-dimensional; (ii) when a circuit with an edge label exists, it leverages the universal cover of the graph to produce an analogous representation and extract an irreducible infinite-dimensional quotient. A third, concrete tree-graph example (isomorphic to ) demonstrates the method in a different combinatorial setting. An appendix sketches a general argument showing that every infinite non-affine Coxeter group has finite-index subgroups surjecting onto a non-abelian free group, enabling an induced irreducible infinite-dimensional representation. Altogether, the work provides explicit constructions that link graph topology to representation theory and yields new infinite-dimensional irreducible representations for broad classes of Coxeter groups.

Abstract

A Coxeter group admits infinite-dimensional irreducible complex representations if and only if it is not finite or affine. In this paper, we provide a construction of some of those representations for certain Coxeter groups using some topological information of the corresponding Coxeter graphs.
Paper Structure (13 sections, 9 theorems, 27 equations, 6 figures)

This paper contains 13 sections, 9 theorems, 27 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representations of dihedral groups
  4. Graphs and the universal covering
  5. The fundamental group of a graph
  6. Representations via fundamental groups of Coxeter graphs
  7. The construction
  8. The infinite-dimensional irreducible quotient
  9. Representations via universal coverings of Coxeter graphs
  10. The construction
  11. The infinite-dimensional irreducible (sub)quotient
  12. Another example
  13. Appendix: A sketched proof of Theorem \ref{['thm1.1']}

Key Result

Theorem 1.1

All irreducible complex representations of $W$ are of finite dimension if and only if $W$ is a finite group or an affine Weyl group.

Figures (6)

  • Figure 1: The representation $\rho_k: D_m \to \operatorname{GL}(\mathbb{C} \beta_r \oplus \mathbb{C} \beta_t)$
  • Figure 2: The edge $\{s_1^\prime, s_2^\prime\}$
  • Figure 4: The vertices $s^\prime$ and $t^\prime$
  • Figure 5: The vertices $s^\prime, t^\prime, s^{\prime\prime}$ and $t^{\prime\prime}$
  • Figure 6: A universal covering
  • ...and 1 more figures

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4: Munkres00
  • Remark 2.5
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 9 more