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A Relationship Between Character Values Of Wreath Products And The Symmetric Group

Rijubrata Kundu, Papi Ray

TL;DR

The paper extends the known connections between irreducible characters of wreath products $G\wr S_n$ and symmetric groups $S_{rn}$, building on Lübeck-Prasad and Roichman-Adin. It employs a wreath-product Murnaghan-Nakayama rule, together with the Roichman-Adin sign correspondence, to derive a new relation for characters evaluated on diagonal elements $(a,\dots,a,\pi)$ when $G$ is abelian of order $r$. The main result expresses $\psi_{\lambda}(\xi^i,\dots,\xi^i;\mu)$ as a product of a root-of-unity $\xi^k$, a sign factor $\mathrm{sign}_r(\hat{\lambda})$, and the symmetric-group character $\chi_{\hat{\lambda}}(w_{r\mu})$, tying wreath-product characters directly to $S_{rn}$-characters. A further extension generalizes the result to any finite abelian group $G$ (not just cyclic), via the $r$-core/quotient framework, broadening the combinatorial bridge between wreath-product representations and symmetric-group theory. Overall, the work deepens the interplay between diagonal character values in wreath products and classical symmetric-group values, with clear implications for combinatorial and representation-theoretic computations.

Abstract

A relation between certain irreducible character values of the hyperoctahedral group $B_n$ ($\mathbb{Z}/2\mathbb{Z} \wr S_n$) and the symmetric group $S_{2n}$ was proved by F. Lübeck and D. Prasad in 2021. Their proof is algebraic in nature and uses Lie theory. Using combinatorial methods, R. Adin and Y. Roichman proved a similar relation between certain character values of $G\wr S_n$ and $S_{rn}$, where $G$ is an abelian group of order $r$ (generalizing the result of Lübeck-Prasad). Using their result, we prove yet another relation between certain irreducible character values of $G\wr S_n$ and $S_{rn}$, where $G$ is an abelian group of order $r$.

A Relationship Between Character Values Of Wreath Products And The Symmetric Group

TL;DR

The paper extends the known connections between irreducible characters of wreath products and symmetric groups , building on Lübeck-Prasad and Roichman-Adin. It employs a wreath-product Murnaghan-Nakayama rule, together with the Roichman-Adin sign correspondence, to derive a new relation for characters evaluated on diagonal elements when is abelian of order . The main result expresses as a product of a root-of-unity , a sign factor , and the symmetric-group character , tying wreath-product characters directly to -characters. A further extension generalizes the result to any finite abelian group (not just cyclic), via the -core/quotient framework, broadening the combinatorial bridge between wreath-product representations and symmetric-group theory. Overall, the work deepens the interplay between diagonal character values in wreath products and classical symmetric-group values, with clear implications for combinatorial and representation-theoretic computations.

Abstract

A relation between certain irreducible character values of the hyperoctahedral group () and the symmetric group was proved by F. Lübeck and D. Prasad in 2021. Their proof is algebraic in nature and uses Lie theory. Using combinatorial methods, R. Adin and Y. Roichman proved a similar relation between certain character values of and , where is an abelian group of order (generalizing the result of Lübeck-Prasad). Using their result, we prove yet another relation between certain irreducible character values of and , where is an abelian group of order .
Paper Structure (7 sections, 5 theorems, 13 equations)

This paper contains 7 sections, 5 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conjugacy classes of $G \wr S_n$
  4. Complex irreducible characters of $G\wr S_n$
  5. Murnaghan-Nakayama rule for irreducible characters of wreath products
  6. The Theorem of Roichman-Adin
  7. The main theorem

Key Result

Theorem 2.2

st Let $G$ be a finite group and $\mathrm{Irr}(G)=\{\chi_0,\cdots,\chi_{r-1}\}$ be a labelling of the irreducible characters of $G$. Let $\psi_{\lambda}$ denote the irreducible character of $G\wr S_n$ indexed by an $r$-partite partition $\lambda$ (with respect to the labelling of $\mathrm{Irr}(G)$). where $\mu=(\mu_1,\cdots,\mu_t)$ and $\mathrm{BST}(\lambda,\mu)$ is the set of all border-strip tab

Theorems & Definitions (14)

  • Example 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • proof
  • Example 4.2
  • Example 4.3
  • ...and 4 more