Algorithmic construction of representations of finite solvable groups

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$p_{i}${}^{\mathrm{\prime }}sareprimes,therealwaysexistsasubnormalseries:$\langle {e} \rangle = G_{o} < G_{1} < \dots < G_{n} = G$suchthat$G_{i}/G_{i-1}$isisomorphictoacyclicgroupoforder$p_{i}$,$i = 1,2,\dots,n$. 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.