Tensor products of irreducible representations of the group G = GL(3,q)

TL;DR

This paper characterizes tensor products of irreducible representations of GL(3,q) using induced and Gelfand-Graev representations, extending classical results and providing decomposition methods, with implications for broader GL(n,q) cases.

Contribution

It offers a new description of tensor products of irreducible representations of GL(3,q) in terms of induced and Gelfand-Graev representations, extending known conjectures and classical results.

Findings

Provides explicit decomposition methods for tensor products.

Extends MacDonald's conjectures to GL(3,q).

Suggests conjectures for GL(n,q).

Abstract

We describe the tensor products of two irreducible linear complex representations of the finite general linear group G = GL(3,q) in terms of induced representations by linear characters of maximal torii and also in terms of Gelfand-Graev representations. Our results include MacDonald's conjectures for G and at the same time they are extensions to G of finite counterparts to classical results on tensor products of holomorphic and anti-holomorphic representations of the group SL(2, R). Moreover they provide an easy way to decompose these tensor products, with the help of Frobenius reciprocity. We also state some conjectures for the general case of GL(n,q).