Representations and primitive central idempotents of a finite solvable group

TL;DR

This paper introduces an inductive approach to construct irreducible representations and primitive central idempotents of finite solvable groups using a special 'long presentation', extending to abelian groups over various fields.

Contribution

It provides a new inductive method for constructing irreducible representations and primitive central idempotents of finite solvable and abelian groups based on a novel 'long presentation'.

Findings

Constructs irreducible representations of finite solvable groups inductively.

Computes primitive central idempotents of complex group algebras for solvable and abelian groups.

Provides systematic methods applicable over fields of characteristic zero or prime to the group order.

Abstract

Let $G$ be a finite solvable group. Then $G$ always has a useful presentation, which we call a "long presentation". Using a "long presentation" of $G$, we present an inductive method of constructing the irreducible representations of $G$ over $\mathbb{C}$ and computing the primitive central idempotents of the complex group algebra $\mathbb{C}[G]$. For a finite abelian group, we present a systematic method of constructing the irreducible representations over a field of characteristic either $0$ or prime to order of the group and also a systematic method of computing the primitive central idempotents of the semisimple abelian group algebra.