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z-Classes in Finite Coxeter Groups

Dilpreet Kaur, Uday Bhaskar Sharma

TL;DR

This work provides a complete enumeration of z-classes for finite Coxeter groups by reducing to irreducible types. It proves that z-classes multiply over direct products, and develops explicit formulas for types $B_n/C_n$ via the wreath product $C_2\wr S_n$, and for type $D_n$ through parity-restricted partitions. The paper further analyzes dihedral groups and compiles z-class counts for exceptional types using GAP/SLA, thereby delivering a comprehensive map of centralizer-structure classes across all finite Coxeter groups and answering a prior question in the literature. The results yield precise combinatorial counts and structural descriptions that facilitate both theoretical understanding and computational verification. Overall, the methods connect centralizer centers, signed partitions, and wreath-product representations to produce exact z-class enumerations with broad applicability in representation theory and group theory.

Abstract

In this paper, we give the enumeration of z-classes in finite Coxeter groups.

z-Classes in Finite Coxeter Groups

TL;DR

This work provides a complete enumeration of z-classes for finite Coxeter groups by reducing to irreducible types. It proves that z-classes multiply over direct products, and develops explicit formulas for types via the wreath product , and for type through parity-restricted partitions. The paper further analyzes dihedral groups and compiles z-class counts for exceptional types using GAP/SLA, thereby delivering a comprehensive map of centralizer-structure classes across all finite Coxeter groups and answering a prior question in the literature. The results yield precise combinatorial counts and structural descriptions that facilitate both theoretical understanding and computational verification. Overall, the methods connect centralizer centers, signed partitions, and wreath-product representations to produce exact z-class enumerations with broad applicability in representation theory and group theory.

Abstract

In this paper, we give the enumeration of z-classes in finite Coxeter groups.
Paper Structure (12 sections, 25 theorems, 54 equations, 1 table)

This paper contains 12 sections, 25 theorems, 54 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $z$-classes in $C_n$ and $B_n$
  4. Conjugacy classes in $C_2 \wr S_n$
  5. Centralizers in $C_2 \wr S_n$
  6. Proof of Theorem \ref{['ThMain1']}
  7. $z$-classes in $D_n$
  8. Conjugacy classes in $D_n$ :
  9. $z$-classes in $D_n,$ when $n$ is odd :
  10. $z$-classes in $D_n,$ when $n$ is even :
  11. Proof of Theorem \ref{['DMain2']} :
  12. $z$-classes of Coxeter groups of exceptional types

Key Result

Theorem 1.1

The total number of $z$-classes in $C_2 \wr S_n$ is: where ${\bm {\lambda}} = k_1^{l_1}\cdots k_u^{l_u}m_1^{w_1}\cdots m_v^{w_v}$ is a partition of $n$, with $1 \leq k_1 < k_2 < \cdots < k_u \leq n$ being odd positive integers and $2\leq m_1 < m_2 < \cdots <m_v \leq n$ being even positive integers.

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 49 more