Algorithmic construction of representations of finite solvable groups

TL;DR

This thesis presents algorithms for constructing irreducible matrix representations of finite solvable groups using generators, primitive idempotents, and algebraic decompositions, with specific methods for abelian groups.

Contribution

It introduces a systematic algorithmic approach for constructing representations of finite solvable groups based on a long system of generators and algebraic idempotents.

Findings

Algorithm for irreducible representation construction over complex numbers.

Explicit expressions for primitive central idempotents of abelian groups.

Systematic computation of primitive idempotents and Wedderburn decomposition.

Abstract

The dominant theme of this thesis is the construction of matrix representations of finite solvable groups using a suitable system of generators. For a finite solvable group $G$ of order $N = p_{1}p_{2}\dots p_{n}$, where $p_{i}$'s are primes, there always exists a subnormal series: $\langle {e} \rangle = G_{o} < G_{1} < \dots < G_{n} = G$ such that $G_{i}/G_{i-1}$ is isomorphic to a cyclic group of order $p_{i}$, $i = 1,2,\dots,n$. Associated with this series, there exists a system of generators consisting $n$ elements $x_{1}, x_{2}, \dots, x_{n}$ (say), such that $G_{i} = \langle x_{1}, x_{2}, \dots, x_{i} \rangle$, $i = 1,2,\dots,n$, which is called a "long system of generators". In terms of this system of generators and conjugacy class sum of $x_{i}$ in $G_{i}$, $i = 1,2, \dots, n$, we present an algorithm for constructing the irreducible matrix representations of $G$ over $\mathbb{C}$ within the group algebra $\mathbb{C}[G]$. This algorithmic construction needs the knowledge of primitive central idempotents, a well defined set of primitive (not necessarily central) idempotents and the "diagonal subalgebra" of $\mathbb{C}[G]$. In terms of this system of generators, we give simple expressions for the primitive central idempotents, a well defined system of primitive (not necessarily central) idempotents and a convenient set of generators of the "diagonal subalgebra" of $\mathbb{C}[G]$. For a finite abelian group, we present an algorithm for constructing the inequivalent irreducible matrix representations over a field of characteristic $0$ or prime to the order of the group and a systematic way of computing the primitive central idempotents of the group algebra. Besides that, we give simple expressions of the primitive central idempotents of the rational group algebra of a finite abelian group using a "long presentation" and it's Wedderburn decomposition.