#### Vol. 14, No. 1, 2020

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Stability in the homology of unipotent groups

### Andrew Putman, Steven V Sam and Andrew Snowden

Vol. 14 (2020), No. 1, 119–154
DOI: 10.2140/ant.2020.14.119
##### Abstract

Let $R$ be a (not necessarily commutative) ring whose additive group is finitely generated and let ${U}_{n}\left(R\right)\subset {GL}_{n}\left(R\right)$ be the group of upper-triangular unipotent matrices over $R$. We study how the homology groups of ${U}_{n}\left(R\right)$ 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 ${U}_{n}\left(R\right)$ over a ring $k$ that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule $\left[n\right]↦{H}_{i}\left({U}_{n}\left(R\right),{M}_{n}\right)$ defines a finitely generated $OI$-module. As a consequence, if $k$ is a field then $dim{H}_{i}\left({U}_{n}\left(R\right),k\right)$ is eventually equal to a polynomial in $n$. We also prove similar results for the Iwahori subgroups of ${GL}_{n}\left(\mathsc{𝒪}\right)$ for number rings $\mathsc{𝒪}$.

##### Keywords
representation stability, unipotent groups, OI-modules, OVI-modules
Primary: 20J05
Secondary: 16P40
##### Milestones
Received: 19 December 2018
Accepted: 18 August 2019
Published: 15 March 2020
##### Authors
 Andrew Putman Department of Mathematics University of Notre Dame Notre Dame, IN United States Steven V Sam Mathematics Department University of California, San Diego La Jolla, CA United States Andrew Snowden Department of Mathematics University of Michigan Ann Arbor, MI United States