On Representations of General Linear Groups over Principal Ideal Local Rings of Length Two

TL;DR

This paper investigates the structure of irreducible complex representations of general linear groups over principal ideal local rings of length two, establishing a canonical correspondence and explicitly constructing representations for small cases.

Contribution

It introduces a canonical correspondence between irreducible representations across these groups and explicitly constructs all such representations for orders three and four.

Findings

Constructed a canonical correspondence preserving dimensions.

Explicitly constructed all irreducible representations for orders three and four.

Showed the complexity of the problem parallels that of arbitrary length rings.

Abstract

We study the irreducible complex representations of general linear groups over principal ideal local rings of length two with a fixed finite residue field. We construct a canonical correspondence between the irreducible representations of all such groups which preserves dimensions. For general linear groups of order three and four over these rings, we construct all the irreducible representations. We show that the the problem of constructing all the irreducible representations of all general linear groups over these rings is not easier than the problem of constructing all the irreducible representations of the general linear groups over principal ideal local rings of arbitrary length in the function field case.