Detecting infinitely many semisimple representations in a fixed finite dimension

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→Mn(k′), where k′ 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.