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

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

representation stability, unipotent groups, OI-modules, OVI-modules
Mathematical Subject Classification 2010
Primary: 20J05
Secondary: 16P40
Received: 19 December 2018
Accepted: 18 August 2019
Published: 15 March 2020
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