The Defining Characteristic Case of the Representations of \(\textrm{GL}_{n}\) and \(\textrm{SL}_{n}\) Over Principal Ideal Local 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 \(\textrm{GL}_{n}\) and \(\textrm{SL}_{n}\) over the principal ideal local rings \(W_{r}(\mathbb {F}_{q})\) and \(\mathbb {F}_{q}[t]/t^{r}\). Let \({\textbf{G}}\) be either \(\textrm{GL}_{n}\) or \(\textrm{SL}_{n}\) and F a perfect field of characteristic p, we prove that for most p the group algebras \(F[{\textbf{G}}(W_{r}(\mathbb {F}_{q}))]\) and \(F[{\textbf{G}}(\mathbb {F}_{q}[t]/t^{r})]\) are not stably equivalent of Morita type. Thus, the group algebras \(F[{\textbf{G}}(W_{r}(\mathbb {F}_{q}))]\) and \(F[{\textbf{G}}(\mathbb {F}_{q}[t]/t^{r})]\) are not isomorphic in the defining characteristic case.