Algebra & Number Theory

Stability in the homology of unipotent groups

Andrew Putman, Steven V Sam, and Andrew Snowden

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Let R be a (not necessarily commutative) ring whose additive group is finitely generated and let U n ( R ) 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 ] 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 .

Article information

Algebra Number Theory, Volume 14, Number 1 (2020), 119-154.

Received: 19 December 2018
Accepted: 18 August 2019
First available in Project Euclid: 7 April 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20J05: Homological methods in group theory
Secondary: 16P40: Noetherian rings and modules

representation stability unipotent groups OI-modules OVI-modules


Putman, Andrew; Sam, Steven V; Snowden, Andrew. Stability in the homology of unipotent groups. Algebra Number Theory 14 (2020), no. 1, 119--154. doi:10.2140/ant.2020.14.119.

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