Representations of $\mathrm{SL}_{n}$ over finite local rings of length two

TL;DR

This paper compares the irreducible representation counts of special linear groups over two related finite local rings of length two, revealing they have identical counts when the characteristic divides the dimension.

Contribution

It establishes a correspondence in the number of irreducible representations between groups over truncated polynomial rings and Witt vector rings of length two under certain divisibility conditions.

Findings

Groups have the same number of irreducible representations of each dimension when p divides n.

Provides insight into the representation theory of linear groups over finite local rings.

Connects algebraic structures over different but related rings.

Abstract

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 $d$.