Representations of $\mathrm{GL}_N$ over finite local principal ideal rings - an overview

TL;DR

This paper surveys the representation theory of the general linear group over finite local principal ideal rings, focusing on regular representations and recent construction methods, summarizing key results and developments in the field.

Contribution

It provides a comprehensive overview of recent advances in constructing regular representations of al_N over finite local principal ideal rings, highlighting new methods and generalizations.

Findings

Summarizes key results of Shintani and Hill.

Details recent constructions by Krakovski--Onn--Singla and Stasinski--Stevens.

Highlights the role of Clifford theory in representation construction.

Abstract

We give a survey of the representation theory of $\mathrm{GL}_N$ over finite local principal ideal rings via Clifford theory, with an emphasis on the construction of regular representations. We review results of Shintani and Hill, and the generalisation of Takase. We then summarise the main features, with some details but without proofs, of the recent constructions of regular representations due to Krakovski--Onn--Singla and Stasinski--Stevens, respectively.