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

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.