Wreath determinants for group-subgroup pairs

TL;DR

This paper introduces a generalized invariant called wreath determinant for finite group-subgroup pairs, extending group determinants by incorporating wreath products and representation theory, with specific focus on abelian groups and examples involving non-abelian pairs.

Contribution

It defines a new invariant for group-subgroup pairs using wreath determinants, linking group theory, matrix invariants, and symmetric group representations.

Findings

Defined the invariant $ heta(G,H)$ for finite groups and subgroups.

Explored properties of wreath determinants in abelian and non-abelian cases.

Provided examples illustrating the application of the invariant.

Abstract

The aim of the present paper is to generalize the notion of the group determinants for finite groups. For a finite group $G$ of order $kn$ and its subgroup $H$ of order $n$, one may define an $n$ by $kn$ matrix $X=(x_{hg^{-1}})_{h\in H,g\in G}$, where $x_g$ ($g\in G$) are indeterminates indexed by the elements in $G$. Then, we define an invariant $\Theta(G,H)$ for a given pair $(G,H)$ by the $k$-wreath determinant of the matrix $X$, where $k$ is the index of $H$ in $G$. The $k$-wreath determinant of $n$ by $kn$ matrix is a relative invariant of the left action by the general linear group of order $k$ and right action by the wreath product of two symmetric groups of order $k$ and $n$. Since the definition of $\Theta(G,H)$ is ordering-sensitive, representation theory of symmetric groups are naturally involved. In this paper, we treat abelian groups with a special choice of indeterminates and give various examples of non-abelian group-subgroup pairs.