Irreducible decomposition and calculating of multiplicity of the symmetric and exterior powers representation of finite groups

TL;DR

This paper develops methods to decompose symmetric and exterior powers of finite group representations into irreducible components using character theory and pre-lambda-ring techniques.

Contribution

It introduces a novel approach for calculating multiplicities of irreducible components in symmetric and exterior powers of finite group representations.

Findings

Derived formulas for multiplicities of irreducible components

Applied character theory and pre-lambda-ring methods

Provided explicit calculations for specific representations

Abstract

In this paper we consider symmetric powers representation and exterior powers representation of finite groups, which generated by the representation which has finite dimension over the complex field. We calculate the multiplicity of irreducible component of two representations of some representation by using a character theory of representation and a pre-lambda-ring, for example, the regular representation.