On the representation ring of the polynomial algebra over a perfect field

TL;DR

This paper describes the structure of the representation ring of polynomial algebras over perfect fields, providing explicit cases for real closed and algebraically closed fields, and extends the discussion to quiver representations, including extended Dynkin types.

Contribution

It offers a general description of the representation ring for polynomial algebras over perfect fields and extends the analysis to quiver representations, including explicit cases.

Findings

Explicit description of the representation ring over perfect fields.

Special cases for real closed and algebraically closed fields.

Representation ring of quivers of extended Dynkin type A.

Abstract

We consider the tensor product of modules over the polynomial algebra corresponding to the usual tensor product of linear operators. We present a general description of the representation ring in case the ground field k is perfect. It is made explicit in the special cases when k is real closed respectively algebraically closed. Furthermore, we discuss the generalisation of this problem to representations of quivers. In particular the representation ring of quivers of extended Dynkin type A is provided.