Stability in the homology of unipotent groups

Putman, Andrew; Sam, Steven V; Snowden, Andrew

doi:10.2140/ant.2020.14.119

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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 $U_{n} (R) \subset {G L}_{n} (R)$ 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 $k$ that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule $[n] \mapsto H_{i} (U_{n} (R), M_{n})$ defines a finitely generated $O I$ -module. As a consequence, if $k$ is a field then $dim H_{i} (U_{n} (R), k)$ is eventually equal to a polynomial in $n$ . We also prove similar results for the Iwahori subgroups of ${G L}_{n} (𝒪)$ for number rings $𝒪$ .

Keywords

representation stability, unipotent groups, OI-modules, OVI-modules

Mathematical Subject Classification 2010

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

Vol. 14, No. 1, 2020