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$E_\infty$-cells and general linear groups of finite fields

Soren Galatius, Alexander Kupers, Oscar Randal-Williams

TL;DR

The paper develops a novel $E_ abla$-algebraic framework to study the mod $p$ homology of ${ m GL}_n(f F_q)$ and establishes new homological stability ranges by constructing CW approximations to classifying spaces in the $E_\infty$-setting, guided by computations with the $E_1$-split Steinberg module. It introduces the $E_1$-splitting complexes and relative Tits buildings to compute $E_1$-homology and then transfers the vanishing results to $E_ abla$-homology via spectral sequences, yielding explicit vanishing lines such as $d< n+r(p-1)-2$ for $q eq 2$ and $d< frac{2}{3}(n-1)$ for $q=2$. The authors resolve Milgram–Priddy’s question by showing the determinant-type class $ ext{det}_3$ lies in the image of restriction from ${ m GL}_6(f F_2)$ to appropriate subgroups, and they provide a comprehensive computation of the Steinberg module homology with coefficients in $f F_ ell$ for $ ell eq p$, including a description via exterior and divided-power algebras. Overall, the work advances the understanding of stability phenomena for linear groups over finite fields and connects representation theory, poset topology, and operadic homology in a coherent framework with concrete computations.

Abstract

We prove new homological stability results for general linear groups over finite fields. These results are obtained by constructing CW approximations to the classifying spaces of these groups, in the category of $E_\infty$-algebras, guided by computations of homology with coefficients in the $E_1$-split Steinberg module.

$E_\infty$-cells and general linear groups of finite fields

TL;DR

The paper develops a novel -algebraic framework to study the mod homology of and establishes new homological stability ranges by constructing CW approximations to classifying spaces in the -setting, guided by computations with the -split Steinberg module. It introduces the -splitting complexes and relative Tits buildings to compute -homology and then transfers the vanishing results to -homology via spectral sequences, yielding explicit vanishing lines such as for and for . The authors resolve Milgram–Priddy’s question by showing the determinant-type class lies in the image of restriction from to appropriate subgroups, and they provide a comprehensive computation of the Steinberg module homology with coefficients in for , including a description via exterior and divided-power algebras. Overall, the work advances the understanding of stability phenomena for linear groups over finite fields and connects representation theory, poset topology, and operadic homology in a coherent framework with concrete computations.

Abstract

We prove new homological stability results for general linear groups over finite fields. These results are obtained by constructing CW approximations to the classifying spaces of these groups, in the category of -algebras, guided by computations of homology with coefficients in the -split Steinberg module.

Paper Structure

This paper contains 27 sections, 26 theorems, 116 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Homological stability
  3. A question of Milgram and Priddy
  4. Homology of Steinberg modules
  5. Recollections on a homology theory for $E_\infty$-algebras
  6. Glossary and some notation from e2cellsIv3
  7. General linear groups as an $E_\infty$-algebra
  8. Construction of $\bold{R}$
  9. Tits buildings and relative Tits buildings
  10. $E_1$-splitting complexes
  11. Computing $E_1$-homology of general linear groups
  12. Poset techniques
  13. The statement
  14. The base cases
  15. The first reduction
  16. ...and 12 more sections

Key Result

Theorem A

If $q = p^r \neq 2$, then

Figures (4)

  • Figure 1: An element of $K$ as in the proof of \ref{['prop:second-reduction-general']} with $n=5$, $w=2$, and $V=\langle e_4, e_5\rangle$.
  • Figure 2: An element of $K'$ when $n=5$ and $L = \langle e_1\rangle$.
  • Figure 3: The additive generators of the $\mathbb{F}_2$-homology of ${\rm GL}_n(\mathbb{F}_2)$ for low degree and low rank.
  • Figure 4: The additive generators of $L$ in the range $n \leq 6$, $d \leq 5$, with filtration degree suppressed.

Theorems & Definitions (61)

  • Theorem A
  • Theorem B
  • Corollary C
  • Lemma 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • Lemma 3.8: Quillen
  • ...and 51 more