Rational Representations and Rational Group Algebra of VZ p-groups

TL;DR

This paper classifies all irreducible rational matrix representations of VZ p-groups and small p-groups, providing explicit formulas for their group algebra decompositions, advancing understanding of their algebraic structure.

Contribution

It explicitly determines all inequivalent irreducible rational matrix representations of p-groups of order up to p^4 and derives combinatorial formulas for their Wedderburn decompositions.

Findings

Complete classification of irreducible rational representations for small p-groups.

Explicit formulas for Wedderburn decompositions of rational group algebras.

Simplified process for analyzing algebraic structures of VZ p-groups.

Abstract

In this article, we study rational matrix representations of VZ $p$-groups ($p$ is any prime). Utilizing our findings on VZ $p$-groups, we explicitly obtain all inequivalent irreducible rational matrix representations of all $p$-groups of order $\leq p^4$. Furthermore, we establish combinatorial formulas to determine the Wedderburn decompositions of rational group algebras for VZ $p$-groups and all $p$-groups of order $\leq p^4$, ensuring simplicity in the process.