Representation Stability and Finite Orthogonal Groups

TL;DR

This paper establishes stability and homological stability results for orthogonal groups over finite commutative rings with 2 invertible, using categorical and algebraic methods to describe their asymptotic structure.

Contribution

It introduces a new categorical framework and proves Noetherianity and stability theorems for orthogonal groups over finite rings, extending previous results.

Findings

Proves stability results for orthogonal groups over finite rings.

Establishes homological stability theorems with untwisted and twisted coefficients.

Provides an asymptotic structure theorem for these groups.

Abstract

In this paper, we prove stability results about orthogonal groups over finite commutative rings where 2 is a unit. Inspired by Putman and Sam (2017), we construct a category $\mathbf{OrI}(R)$ and prove a Noetherianity theorem for the category of $\mathbf{OrI}(R)$-modules. This implies an asymptotic structure theorem for orthogonal groups. In addition, we show general homological stability theorems for orthogonal groups, with both untwisted and twisted coefficients, partially generalizing a result of Charney (1987).