Stability in the homology of unipotent groups

TL;DR

This paper investigates the homology of unipotent groups over rings and demonstrates that their homology groups exhibit representation stability, eventually becoming polynomial in dimension for large n.

Contribution

It establishes that under certain conditions, the homology groups form finitely generated modules, leading to polynomial growth of their dimensions in large n.

Findings

Homology groups of unipotent groups stabilize as n grows.

Dimensions of homology groups are eventually polynomial in n.

Results extend to Iwahori subgroups of general linear groups over number rings.

Abstract

Let $R$ be a (not necessarily commutative) ring whose additive group is finitely generated and let $U_n(R) \subset GL_n(R)$ be the group of upper-triangular unipotent matrices over $R$. We study how the homology groups of $U_n(R)$ vary with $n$ from the point of view of representation stability. Our main theorem asserts that if for each $n$ we have representations $M_n$ of $U_n(R)$ over a ring $\mathbf{k}$ that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule $[n] \mapsto \widetilde{H}_i(U_n(R),M_n)$ defines a finitely generated OI-module. As a consequence, if $\mathbf{k}$ is a field then $dim \widetilde{H}_i(U_n(R),\mathbf{k})$ is eventually equal to a polynomial in $n$. We also prove similar results for the Iwahori subgroups of $GL_n(\mathcal{O})$ for number rings $\mathcal{O}$.