Work

Double Steinberg coinvariants for special linear groups

Year: 2026

Type: preprint

Abstract: For a field $F$ we study the coinvariants for the $SL_n(F)$-action on the double Steinberg module $St_n(F) \otimes St_n(F)$ 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 $\mathbb{Z}[F^\times]$-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 $E_\infty$-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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.39991

Authors: Abdelnaim, Tatiana, Chan, David, Kupers, Alexander, Sroka, Robin J., Scalamandre, Matthew

Institutions: University of Toronto, University of Toronto Scarborough, University of Oklahoma, Michigan State University, University of Münster

Language: English

Topic: Homotopy and Cohomology in Algebraic Topology

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Vanderbilt University, University of Pennsylvania

Award: Patterns in the Cohomology of Linear Groups

OpenAlex ID: W7215062603

Cites: 3

Cited by:

Related to: 10