Resolutions of the Steinberg module for GL(n)

Avner Ash; Paul E. Gunnells; Mark McConnell

doi:10.1016/j.jalgebra.2011.09.018

Resolutions of the Steinberg module for GL(n)

Avner Ash, Paul E. Gunnells, Mark McConnell

Research output: Contribution to journal › Article › peer-review

15 Scopus citations

Abstract

We give several resolutions of the Steinberg representation St_n for the general linear group over a principal ideal domain, in particular over Z{double-struck}. We compare them, and use these results to prove that the computations in Avner Ash et al. (2011) [AGM11] are definitive. In particular, in Avner Ash et al. (2011) [AGM11] we use two complexes to compute certain cohomology groups of congruence subgroups of SL(4,Z{double-struck}). One complex is based on Voronoi's polyhedral decomposition of the symmetric space for SL(n,R{double-struck}), whereas the other is a larger complex that has an action of the Hecke operators. We prove that both complexes allow us to compute the relevant cohomology groups, and that the use of the Voronoi complex does not introduce any spurious Hecke eigenclasses.

Original language	English (US)
Pages (from-to)	380-390
Number of pages	11
Journal	Journal of Algebra
Volume	349
Issue number	1
DOIs	https://doi.org/10.1016/j.jalgebra.2011.09.018
State	Published - Jan 1 2012
Externally published	Yes

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

Keywords

Cohomology of arithmetic groups
Modular symbols
Steinberg module
Voronoi complex

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10.1016/j.jalgebra.2011.09.018

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title = "Resolutions of the Steinberg module for GL(n)",

abstract = "We give several resolutions of the Steinberg representation Stn for the general linear group over a principal ideal domain, in particular over Z{double-struck}. We compare them, and use these results to prove that the computations in Avner Ash et al. (2011) [AGM11] are definitive. In particular, in Avner Ash et al. (2011) [AGM11] we use two complexes to compute certain cohomology groups of congruence subgroups of SL(4,Z{double-struck}). One complex is based on Voronoi's polyhedral decomposition of the symmetric space for SL(n,R{double-struck}), whereas the other is a larger complex that has an action of the Hecke operators. We prove that both complexes allow us to compute the relevant cohomology groups, and that the use of the Voronoi complex does not introduce any spurious Hecke eigenclasses.",

keywords = "Cohomology of arithmetic groups, Modular symbols, Steinberg module, Voronoi complex",

author = "Avner Ash and Gunnells, {Paul E.} and Mark McConnell",

note = "Funding Information: ✩ A.A. wishes to thank the National Science Foundation for support of this research through NSF grant DMS-0455240, and also the NSA through grant H98230-09-1-0050. This manuscript is submitted for publication with the understanding that the United States government is authorized to produce and distribute reprints. P.G. wishes to thank the National Science Foundation for support of this research through NSF grant DMS-0801214. * Corresponding author. E-mail addresses: [email protected] (A. Ash), [email protected] (P.E. Gunnells), [email protected] (M. McConnell).",

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AU - Gunnells, Paul E.

AU - McConnell, Mark

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N2 - We give several resolutions of the Steinberg representation Stn for the general linear group over a principal ideal domain, in particular over Z{double-struck}. We compare them, and use these results to prove that the computations in Avner Ash et al. (2011) [AGM11] are definitive. In particular, in Avner Ash et al. (2011) [AGM11] we use two complexes to compute certain cohomology groups of congruence subgroups of SL(4,Z{double-struck}). One complex is based on Voronoi's polyhedral decomposition of the symmetric space for SL(n,R{double-struck}), whereas the other is a larger complex that has an action of the Hecke operators. We prove that both complexes allow us to compute the relevant cohomology groups, and that the use of the Voronoi complex does not introduce any spurious Hecke eigenclasses.

AB - We give several resolutions of the Steinberg representation Stn for the general linear group over a principal ideal domain, in particular over Z{double-struck}. We compare them, and use these results to prove that the computations in Avner Ash et al. (2011) [AGM11] are definitive. In particular, in Avner Ash et al. (2011) [AGM11] we use two complexes to compute certain cohomology groups of congruence subgroups of SL(4,Z{double-struck}). One complex is based on Voronoi's polyhedral decomposition of the symmetric space for SL(n,R{double-struck}), whereas the other is a larger complex that has an action of the Hecke operators. We prove that both complexes allow us to compute the relevant cohomology groups, and that the use of the Voronoi complex does not introduce any spurious Hecke eigenclasses.

KW - Cohomology of arithmetic groups

KW - Modular symbols

KW - Steinberg module

KW - Voronoi complex

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Resolutions of the Steinberg module for GL(n)

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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