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On exterior powers of reflection representations, II

Hongsheng Hu

TL;DR

The paper proves that for any irreducible reflection representation $V$ of a group $W$ with a finite generating set $S$, all exterior powers $\bigwedge^d V$ are irreducible and pairwise non-isomorphic as $W$-modules. It further shows that exterior powers of two non-isomorphic reflection representations remain non-isomorphic (excluding the trivial $d=0$ and top-degree cases), enabling the construction of many pairwise non-isomorphic irreducibles, particularly for Coxeter groups. The approach combines a digraph framework encoding the reflection data with linear-algebraic arguments to control endomorphisms and to build explicit intertwiners between wedge powers. This generalizes Steinberg's theorem beyond Euclidean settings and yields a rich supply of irreducible representations, with applications to affine Weyl groups and generalized geometric representations. The results significantly expand the toolkit for understanding representation theory of Coxeter-type groups through exterior powers and associated combinatorial structures.

Abstract

Let $W$ be a group endowed with a finite set $S$ of generators. A representation $(V,ρ)$ of $W$ is called a reflection representation of $(W,S)$ if $ρ(s)$ is a (generalized) reflection on $V$ for each generator $s \in S$. In this paper, we prove that for any irreducible reflection representation $V$, all the exterior powers $\bigwedge ^d V$, $d = 0, 1, \dots, \dim V$, are irreducible $W$-modules, and they are non-isomorphic to each other. This extends a theorem of R. Steinberg which is stated for Euclidean reflection groups. Moreover, we prove that the exterior powers (except for the 0th and the highest power) of two non-isomorphic reflection representations always give non-isomorphic $W$-modules. This allows us to construct numerous pairwise non-isomorphic irreducible representations for such groups, especially for Coxeter groups.

On exterior powers of reflection representations, II

TL;DR

The paper proves that for any irreducible reflection representation of a group with a finite generating set , all exterior powers are irreducible and pairwise non-isomorphic as -modules. It further shows that exterior powers of two non-isomorphic reflection representations remain non-isomorphic (excluding the trivial and top-degree cases), enabling the construction of many pairwise non-isomorphic irreducibles, particularly for Coxeter groups. The approach combines a digraph framework encoding the reflection data with linear-algebraic arguments to control endomorphisms and to build explicit intertwiners between wedge powers. This generalizes Steinberg's theorem beyond Euclidean settings and yields a rich supply of irreducible representations, with applications to affine Weyl groups and generalized geometric representations. The results significantly expand the toolkit for understanding representation theory of Coxeter-type groups through exterior powers and associated combinatorial structures.

Abstract

Let be a group endowed with a finite set of generators. A representation of is called a reflection representation of if is a (generalized) reflection on for each generator . In this paper, we prove that for any irreducible reflection representation , all the exterior powers , , are irreducible -modules, and they are non-isomorphic to each other. This extends a theorem of R. Steinberg which is stated for Euclidean reflection groups. Moreover, we prove that the exterior powers (except for the 0th and the highest power) of two non-isomorphic reflection representations always give non-isomorphic -modules. This allows us to construct numerous pairwise non-isomorphic irreducible representations for such groups, especially for Coxeter groups.
Paper Structure (16 sections, 32 theorems, 106 equations)

This paper contains 16 sections, 32 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Motivation and application
  4. Outline of this paper
  5. Preliminaries
  6. Reflections and reflection representations
  7. Digraphs
  8. Some numerical lemmas
  9. Exterior powers of reflection representations
  10. Proof of Theorem \ref{['thm-main1']}
  11. Proof of Theorem \ref{['thm-main2']}
  12. A preliminary numerical result
  13. Preliminary results on linear independence of reflection vectors
  14. Coincidence of the associated digraphs
  15. The linear isomorphism f from V1 to V2
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(V,\rho)$ be an $n$-dimensional irreducible reflection representation of $(W,S)$ over a field $\mathbb{F}$ of characteristic 0, with reflection vectors $\alpha_1, \dots, \alpha_k$. Suppose Then the $W$-modules $\{\bigwedge^d V \mid 0 \le d \le n\}$ are irreducible and pairwise non-isomorphic.

Theorems & Definitions (62)

  • Theorem 1.1: Hu23-ext-pow
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Definition 2.1: Hu23-ext-pow
  • Lemma 2.2: Hu23-ext-pow
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 52 more