On representation zeta function of special linear groups over finite principal ideal local rings

Abstract

We show that the group algebras $\mathbb{C}[\text{SL}_3(\mathbb{F}_3[t]/(t^3))]$ and $\mathbb{C}[\text{SL}_3(\mathbb{Z}/27)]$ are not isomorphic, as well as $\mathbb{C}[\text{SL}_4(\mathbb{F}_2[t]/(t^3))]$ and $\mathbb{C}[\text{SL}_4(\mathbb{Z}/8)]$, by computing the number of conjugacy classes in those groups using MAGMA's calculator. Similarly, we reproduce special cases of a recent result by Hassain and Singla, showing that $\mathbb{C}[\text{SL}_2(\mathbb{F}_2[t]/(t^k))]\ncong\mathbb{C}[\text{SL}_2(\mathbb{Z}/2^k)]$ for $3\leq k\leq 8$.