Representations of $\mathrm{SL}_{n}$ over finite local rings of length two
Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$ and let $W_{2}(\mathbb{F}_{q})$ be the ring of Witt vectors of length two over $\mathbb{F}_{q}$. We prove that for any integer $n$ such that $p$ divides $n$, the groups $\mathrm{SL}_{n}(\mathbb{F}_{q}[t]/t^{2})$ and $\mathrm{SL}_{n}(W_{2}(\mathbb{F}_{q}))$ have the same number of irreducible representations of dimension $d$, for each $d$.