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Stiefel-Whitney Classes for Finite Special Linear Groups of Even Rank

Neha Malik, Steven Spallone

TL;DR

This work determines the Stiefel-Whitney classes of orthogonal representations of SL(n,q) with odd rank and odd q by expressing the total SWC w(pi) as a product of Dickson-factor polynomials whose exponents are governed by character values on diagonal elements of order 2. The authors develop a detection framework using diagonal subgroups and elementary abelian 2-groups, relate representations to S_r- and S_{r+1}-invariant constituents, and deploy Dickson invariants to encode the SWCs via symmetric functions. They give explicit computations for SL(3,q) and SL(5,q), including formulas for w4, w8, and in some cases w16, along with obstruction criteria for nonvanishing top SWCs and universal detection results via smaller SL groups. The work also establishes the structure of H^*_{SW}(G) and the Stiefel-Whitney group, and provides universal formulas for early SWCs (notably w4 and w8) applicable for sufficiently large n, thereby facilitating systematic SWC calculations across families SL(n,q).

Abstract

We compute the total Stiefel-Whitney Classes (SWCs) for orthogonal representations of special linear groups $\text{SL}(n,q)$ when $n$ and $q$ are odd. These classes are expressed in terms of character values at diagonal elements of order $2$. We give several consequences, and work out the $4$th SWC explicitly, and the $8$th SWC when the $4$th vanishes.

Stiefel-Whitney Classes for Finite Special Linear Groups of Even Rank

TL;DR

This work determines the Stiefel-Whitney classes of orthogonal representations of SL(n,q) with odd rank and odd q by expressing the total SWC w(pi) as a product of Dickson-factor polynomials whose exponents are governed by character values on diagonal elements of order 2. The authors develop a detection framework using diagonal subgroups and elementary abelian 2-groups, relate representations to S_r- and S_{r+1}-invariant constituents, and deploy Dickson invariants to encode the SWCs via symmetric functions. They give explicit computations for SL(3,q) and SL(5,q), including formulas for w4, w8, and in some cases w16, along with obstruction criteria for nonvanishing top SWCs and universal detection results via smaller SL groups. The work also establishes the structure of H^*_{SW}(G) and the Stiefel-Whitney group, and provides universal formulas for early SWCs (notably w4 and w8) applicable for sufficiently large n, thereby facilitating systematic SWC calculations across families SL(n,q).

Abstract

We compute the total Stiefel-Whitney Classes (SWCs) for orthogonal representations of special linear groups when and are odd. These classes are expressed in terms of character values at diagonal elements of order . We give several consequences, and work out the th SWC explicitly, and the th SWC when the th vanishes.
Paper Structure (25 sections, 22 theorems, 58 equations)

This paper contains 25 sections, 22 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Representations
  4. Stiefel-Whitney Classes
  5. Detection
  6. Symmetric Functions
  7. Cyclic Groups
  8. Elementary Abelian $2$-groups
  9. Linear Characters
  10. $S_r$-invariant Representations of $E$
  11. $S_{r+1}$-invariant representations of $E$
  12. Dickson Factors
  13. A review of Joshi-Ganguly's work on $\mathop{\mathrm{GL}}\nolimits(n,q)$
  14. Case $q\equiv 3\;(\mathop{\mathrm{mod}}\nolimits\: 4)$
  15. Case $q\equiv 1\;(\mathop{\mathrm{mod}}\nolimits\: 4)$
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\pi$ be an orthogonal representation of $\mathop{\mathrm{SL}}\nolimits(n,q)$ with odd $n \geq 5$. Then $w_4(\pi)=0$ iff $m_1 \equiv m_2 \pmod 4$. Moreover, when $q \equiv 1 \pmod 4$ and $w_4(\pi)=0$, then $w_8(\pi)=0$ iff $m_1 \equiv m_2 \pmod 8$.

Theorems & Definitions (42)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Theorem 2.2: Quillen, Theorem 3
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 32 more