Constructing irreducible representations of finitely presented algebras

Abstract

By combining well-known techniques from both noncommutative algebra and computational commutative algebra, we observe that an algorithmic approach can be applied to the study of irreducible representations of finitely presented algebras. In slightly more detail: Assume that $n$ is a positive integer, that $k$ is a computable field, that $\bar{k}$ denotes the algebraic closure of $k$, and that $M_n(\bar{k})$ denotes the algebra of $n \times n$ matrices with entries in $\bar{k}$. Let $R$ be a finitely presented $k$-algebra. Calculating over $k$, the procedure will (a) decide whether an irreducible representation $R \to M_n(\bar{k})$ exists, and (b) explicitly construct an irreducible representation $R \to M_n(\bar{k})$ if at least one exists. (For (b), it is necessary to assume that $k[x]$ is equipped with a factoring algorithm.) An elementary example is worked through.