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Homological vanishing for the Steinberg representation

Part of: Linear algebraic groups and related topics Connections with homological algebra and category theory

Published online by Cambridge University Press:  29 April 2018

Avner Ash ,
Andrew Putman  and
Steven V Sam
Avner Ash
Affiliation:
Department of Mathematics, Boston College, Chestnut Hill, MA 02467-3806, USA email ashav@bc.edu
Andrew Putman
Affiliation:
Department of Mathematics, University of Notre Dame, 279 Hurley Hall, Notre Dame, IN 46556, USA email andyp@nd.edu
Steven V Sam
Affiliation:
Department of Mathematics, University of Wisconsin, 480 Lincoln Dr., Madison, WI 53706-1325, USA email svs@math.wisc.edu
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Abstract

For a field $\text{k}$, we prove that the $i$th homology of the groups $\operatorname{GL}_{n}(\text{k})$, $\operatorname{SL}_{n}(\text{k})$, $\operatorname{Sp}_{2n}(\text{k})$, $\operatorname{SO}_{n,n}(\text{k})$, and $\operatorname{SO}_{n,n+1}(\text{k})$ with coefficients in their Steinberg representations vanish for $n\geqslant 2i+2$.

Keywords

Steinberg representationhomological stability

MSC classification

Primary: 20J05: Homological methods in group theory 20G10: Cohomology theory
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 6 , June 2018 , pp. 1111 - 1130
Copyright
© The Authors 2018 

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