A Combinatorial Formula for the Wedderburn Decomposition of Rational Group Algebras and the Rational Representations of Ordinary Metacyclic $p$-groups

TL;DR

This paper introduces a combinatorial approach to decompose rational group algebras of metacyclic p-groups, enabling explicit computation of their irreducible rational representations and their degrees.

Contribution

It provides a new combinatorial formula for Wedderburn decomposition and a method to explicitly find all inequivalent irreducible rational matrix representations of metacyclic p-groups.

Findings

Derived a formula for Wedderburn decomposition of rational group algebras

Developed a method to count irreducible rational representations with distinct degrees

Explicitly obtained all inequivalent irreducible rational matrix representations

Abstract

In this article, we present a combinatorial formula for computing the Wedderburn decomposition of the rational group algebra associated with an ordinary metacyclic $p$-group $G$, where $p$ is any prime. We also provide a formula for counting irreducible rational representations of $G$ with distinct degrees and derive a method to explicitly obtain all inequivalent irreducible rational matrix representations of $G$.