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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 16, Issue 1, 1997, pp. 145–164**

**DOI: 10.4171/ZAA/756**

Published online: 1997-03-31

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

Bernold Fiedler^{[1]}(1) Freie Universität Berlin, Germany

Let $I$ be a left ideal of a group ring $\mathbb C[G]$ of a finite group $G$, for which a decomposition $I = \oplus ^m_{k=1} I_k$ 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.

*Keywords: *Group rings, ideal decompositions, primitive orthogonal idempotents, Young symmetrizers, the regular representation of the $S_r$, invariant irreducible subspaces, computer-aided tensor calculations, Ricci calculus

Fiedler Bernold: A Use of Ideal Decomposition in the Computer Algebra of Tensor Expressions. *Z. Anal. Anwend.* 16 (1997), 145-164. doi: 10.4171/ZAA/756