How to compute the Wedderburn decomposition of a finite-dimensional associative algebra. (Q2882824)

Let \(A\) be finite-dimensional algebra over either a finite field or an algebraically closed field. The paper under review surveys various algorithms available to do the following:{\parindent=8mm\begin{itemize}\item[(i)] compute a basis for the radical \(R\) of \(A\);\item[(ii)] compute structure constants for the semisimple quotient \(A/R\);\item[(iii)] compute a basis for the center of \(A/R\) consisting of orthogonal idempotents;\item[(iv)] compute the identity matrices in simple ideals of \(A/R\);\item[(v)] compute an isomorphism of each simple ideal of \(A/R\) with a full matrix algebra;\item[(vi)] compute explicit matrices for irreducible representations of \(A\).NEWLINENEWLINE\end{itemize}} The author illustrates these algorithms on the example of the semigroup algebra of the full partial transformation semigroup \(PT_2\) on a \(2\)-element set.