The Defining Characteristic Case of the Representations of $\mathrm{GL}_{n}$ and $\mathrm{SL}_{n}$ over Principal Ideal Local Rings

TL;DR

This paper investigates the representation theory of general linear and special linear groups over principal ideal local rings, showing that their group algebras are generally not Morita equivalent or isomorphic in the defining characteristic case.

Contribution

It proves that for most primes, the group algebras over these rings are not Morita equivalent or isomorphic, highlighting differences in their representation structures in the defining characteristic case.

Findings

Group algebras are not stably equivalent of Morita type for most p.

Group algebras are not isomorphic in the defining characteristic case.

Results apply to rings of Witt vectors and polynomial quotient rings.

Abstract

Let $W_{r}(\mathbb{F}_{q})$ be the ring of Witt vectors of length $r$ with residue field $\mathbb{F}_{q}$ of characteristic $p$. In this paper, we study the defining characteristic case of the representations of $\mathrm{GL}_{n}$ and $\mathrm{SL}_{n}$ over the principal ideal local rings $W_{r}(\mathbb{F}_{q})$ and $\mathbb{F}_{q}[t]/t^{r}$. Let ${\mathbf{G}}$ be either $\mathrm{GL}_{n}$ or $\mathrm{SL}_{n}$ and $F$ a perfect field of characteristic $p$, we prove that for most $p$ the group algebras $F[{\mathbf{G}}(W_{r}(\mathbb{F}_{q}))]$ and $F[{\mathbf{G}}(\mathbb{F}_{q}[t]/t^{r})]$ are not stably equivalent of Morita type. Thus, the group algebras $F[{\mathbf{G}}(W_{r}(\mathbb{F}_{q}))]$ and $F[{\mathbf{G}}(\mathbb{F}_{q}[t]/t^{r})]$ are not isomorphic in the defining characteristic case.