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Irreducible characters of the generalized symmetric group

Huimin Gao, Naihuan Jing

TL;DR

The paper develops vertex-operator methods to compute irreducible characters of the generalized symmetric group $C_k\\wr S_n$, reformulating the Ariki–Koike Murnaghan-Nakayama rule and deriving a dual iteration for these characters. It expresses character values in terms of colored partitions and colored rim-hook combinatorics, enabling recursive computations and practical SageMath/Zim implementations. A degree formula and a mod k relation to modular characters of $S_{kn}$ are established, with explicit low-rank examples and tables for small $k$ and $n$, providing new links between wreath-product characters and modular representation theory.

Abstract

The paper studies how to compute irreducible characters of the generalized symmetric group $C_k\wr{S}_n$ by iterative algorithms. After reproving the Ariki-Koike version of the Murnaghan-Nakayama rule by vertex algebraic methods, we formulate a new iterative formula for characters of the generalized symmetric group. As applications, we find a numerical relation between the character values of $C_k\wr S_n$ and modular characters of $S_{kn}$.

Irreducible characters of the generalized symmetric group

TL;DR

The paper develops vertex-operator methods to compute irreducible characters of the generalized symmetric group , reformulating the Ariki–Koike Murnaghan-Nakayama rule and deriving a dual iteration for these characters. It expresses character values in terms of colored partitions and colored rim-hook combinatorics, enabling recursive computations and practical SageMath/Zim implementations. A degree formula and a mod k relation to modular characters of are established, with explicit low-rank examples and tables for small and , providing new links between wreath-product characters and modular representation theory.

Abstract

The paper studies how to compute irreducible characters of the generalized symmetric group by iterative algorithms. After reproving the Ariki-Koike version of the Murnaghan-Nakayama rule by vertex algebraic methods, we formulate a new iterative formula for characters of the generalized symmetric group. As applications, we find a numerical relation between the character values of and modular characters of .
Paper Structure (6 sections, 16 theorems, 77 equations, 4 tables)

This paper contains 6 sections, 16 theorems, 77 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Irreducible characters of $C_k\wr S_n$
  3. Two Iterative Formulas
  4. Further examples
  5. The source code for Theorem \ref{['t:MN-gensym']}
  6. Theorem \ref{['t:3.9']}

Key Result

Theorem 1.1

Given colored partitions $\boldsymbol{\lambda}=(\lambda^{(0)},\lambda^{(1)},\cdots,\lambda^{(k-1)})$ and $\boldsymbol{\rho}=(\rho^{(0)},\rho^{(1)},\cdots,\rho^{(k-1)})$ of $n$, where $\rho^{(s)}=(\rho^{(s)}_1,\cdots,\rho^{(s)}_m,\cdots,\rho^{(s)}_{l(s)})$. Then the value of the irreducible character where ${\boldsymbol{\xi_j}}$ runs through all colored $\rho^{(s)}_m$-rim hooks contained in ${\bold

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Theorem 2.2
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Example 3.4
  • ...and 18 more