The Defining Characteristic Case of the Representations of ${\mathrm{G}\mathrm{L}}_n$ and ${\mathrm{S}\mathrm{L}}_n$ over Principal Ideal Local Rings

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 ${\mathrm{G}\mathrm{L}}_n$ and ${\mathrm{S}\mathrm{L}}_n$ over the principal ideal local rings $W_r({\mathbb{F}}_q)$ and ${\mathbb{F}}_q[t]\mathrm{/}{t}^r$. Let $\mathbf{G}$ be either ${\mathrm{G}\mathrm{L}}_n$ or ${\mathrm{S}\mathrm{L}}_n$ 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 $F[\mathbf{G}({\mathbb{F}}_q[t]\mathrm{/}{t}^r)]$ are not stably equivalent of Morita type. Thus, the group algebras $F[\mathbf{G}(W_r({\mathbb{F}}_q))]$ and $F[\mathbf{G}({\mathbb{F}}_q[t]\mathrm{/}{t}^r)]$ are not isomorphic in the defining characteristic case.