Detecting Infinitely Many Semisimple Representations in a Fixed Finite Dimension

Abstract

Let $n$ be a positive integer, and let $k$ be a field (of arbitrary characteristic) accessible to symbolic computation. We describe an algorithmic test for determining whether or not a finitely presented $k$-algebra $R$ has infinitely many equivalence classes of semisimple representations $R\to M_n(k^{\prime })$, where $k^{\prime }$ is the algebraic closure of $k$. The test reduces the problem to computational commutative algebra over $k$, via famous results of Artin, Procesi, and Shirshov. The test is illustrated by explicit examples, with $n=3$.

12 pages, no figures. Revised; to appear in Journal of Algebra (Computational Section)