Divisibility of Character Values of Representations of Coxeter Groups

TL;DR

The paper analyzes the asymptotic divisibility of irreducible character values across infinite families of irreducible Coxeter groups by introducing the statistic $\mathscr{L}(\mathscr{C}(G_n, G_0), g, d)$. It shows that for types $A_n$, $B_n$, and $D_n$ the proportion of irreducible characters with values divisible by a fixed $d$ tends to $1$, while for dihedral groups $I_2(m)$ the limit is computed explicitly in terms of $m$, $l$, and $d$, including 2-adic conditions. The results are established through detailed analyses of conjugacy classes and irreducible representations (notably bipartitions for $\mathbb{B}_n$ and induced representations), together with integrality arguments involving centralizers and explicit character formulas. By handling Hyperoctahedral, Demi-Hyperoctahedral, and Dihedral families, the work provides a unified framework for the divisibility behavior of character values across all infinite Coxeter families, with concrete limits depending on the group family and chosen elements.

Abstract

Let $d$ be a positive integer. We study the proportion of irreducible characters of infinite families of irreducible Coxeter groups whose values evaluated on a fixed element $g$ are divisible by $d$. For Coxeter groups of types $A_n, B_n$ and $D_n$, the proportion tends to $1$ as $n$ approaches infinity. For Dihedral groups, which are Coxeter groups of type $I_2(n)$, we compute the limit of the proportion.

Figures (1)

• Figure 1: The figure shows an embedding of $\Delta_3=P_1P_3P_5$ inside $\Delta_6$.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Lemma 3.1
• Lemma 3.2
• proof
• proof : Proof of \ref{['mt']}
• Theorem 4.1
• Lemma 4.2