Matrix units in the simple components of rational group algebras

TL;DR

The paper presents a practical method for computing matrix units and primitive idempotents in simple components of rational group algebras for certain finite groups, aiding algebraic structure analysis.

Contribution

It introduces an effective computational approach for matrix units in simple components of rational group algebras using generalized strongly monomial characters with prime Schur index.

Findings

Method successfully computes matrix units for specific group classes

Application demonstrated through detailed computational examples

Enhances understanding of algebraic structure of rational group algebras

Abstract

For the rational group algebra $\mathbb{Q}G$ of a finite group $G$, we provide an effective method to compute a complete set of matrix units and, in particular, primitive orthogonal idempotents in a simple component of $\mathbb{Q}G$, which is realized by a generalized strongly monomial character and has a prime Schur index. We also provide some classes of groups $G$ where this method can be successfully applied. The application of the method developed is also illustrated with detailed computations.