A Use of Ideal Decomposition in the Computer Algebra of Tensor Expressions

Abstract

Let $I$ be a left ideal of a group ring $\mathbb{C}[G]$ of a finite group $G$, for which a decomposition $I={\oplus }_{k=1}^m{I}_k$ into minimal left ideals $I_k$ is given. We present an algorithm, which determines a decomposition of the left ideal $I\cdot a,a\in \mathbb{C}[G]$, into minimal left ideals and a corresponding set of primitive orthogonal idempotents by means of a computer. The algorithm is motivated by the computer algebra of tensor expressions. Several aspects of the connection between left ideals of the group ring $\mathbb{C}[S_r]$ of a symmetric group $S_r$, their decomposition and the reduction of tensor expressions are discussed.