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Double Steinberg coinvariants for special linear groups
Abstract
For a field F we study the coinvariants for the SLn(F)-action on the double Steinberg module Stn(F) ⊗ Stn(F) and show they have a rich algebraic structure: for n = 2 it is the Grothendieck-Witt group of F, and for all n they assemble to a graded nonunital ℤ[F×]-algebra, whose rational (underived) indecomposables may be expressed in terms of the augmentation ideal of the Grothendieck-Witt group. We then explain, building on work of Galatius-Kupers-Randal-Williams, that the special linear groups SLn(F) assemble to an E∞-algebra in a suitable functor category, whose E2-homology groups have a vanishing line of slope 2 and on the critical line are given by the double Steinberg coinvariants.
Related
- General linear and Steinberg groups over the Leavitt algebra L𝔽2(1,2)
- Complete mappings stabilizing Sylow $2$-subgroups in central quotients of special linear groups
- Galois coinvariants and fundamental groups of arithmetic schemes
- Affine iquantum groups and Steinberg varieties of type C
- Steinberg module and top-dimensional cohomology for arithmetic Chevalley groups
BibTeX
@misc{abdelnaim2026double,
title = {{Double Steinberg coinvariants for special linear groups}},
author = {Tatiana Abdelnaim and David Chan and Alexander Kupers and Robin J. Sroka and Matthew Scalamandre},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39991},
eprint = {2609.39991},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39991},
}
RIS
TY - UNPB
TI - Double Steinberg coinvariants for special linear groups
AU - Tatiana Abdelnaim
AU - David Chan
AU - Alexander Kupers
AU - Robin J. Sroka
AU - Matthew Scalamandre
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39991
UR - https://arxiv.org/abs/2609.39991
AB - For a field F we study the coinvariants for the SL_n(F)-action on the double Steinberg module St_n(F) ⊗ St_n(F) and show they have a rich algebraic structure: for n = 2 it is the Grothendieck-Witt group of F, and for all n they assemble to a graded nonunital ℤ[F^×]-algebra, whose rational (underived) indecomposables may be expressed in terms of the augmentation ideal of the Grothendieck-Witt group. We then explain, building on work of Galatius-Kupers-Randal-Williams, that the special linear groups SL_n(F) assemble to an E_∞-algebra in a suitable functor category, whose E_2-homology groups have a vanishing line of slope 2 and on the critical line are given by the double Steinberg coinvariants.
ER -