Detecting Infinitely Many Semisimple Representations in a Fixed Finite Dimension

TL;DR

This paper introduces an algorithmic method to determine whether a finitely presented algebra over a field has infinitely many semisimple representations of a fixed dimension, utilizing computational algebra techniques.

Contribution

The paper presents a novel algorithm that reduces the problem to computational commutative algebra, enabling effective detection of infinite semisimple representations.

Findings

Algorithm successfully distinguishes finite vs. infinite semisimple representations.

Illustrative examples demonstrate the algorithm's application for n=3.

Method applies to fields of arbitrary characteristic.

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 $R \to M_n(k')$, 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 $n = 3$.