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Invariants of Finite Orthogonal Groups of Plus Type in Odd Characteristic

H. E. A. Campbell, R. James Shank, David L. Wehlau

Abstract

We describe the rings of invariants for the finite orthogonal groups of plus type in odd characteristic acting on the defining representations. We also describe the invariants of the corresponding Sylow subgroups in the defining characteristic. In both cases we construct minimal algebra generating sets and describe the relations among the generators. Both rings of invariants are shown to be complete intersections and thus are Cohen-Macaulay. We expect the techniques we use will generalise to give a systematic computation for rings of invariants for all of the finite classical groups in odd characteristic.

Invariants of Finite Orthogonal Groups of Plus Type in Odd Characteristic

Abstract

We describe the rings of invariants for the finite orthogonal groups of plus type in odd characteristic acting on the defining representations. We also describe the invariants of the corresponding Sylow subgroups in the defining characteristic. In both cases we construct minimal algebra generating sets and describe the relations among the generators. Both rings of invariants are shown to be complete intersections and thus are Cohen-Macaulay. We expect the techniques we use will generalise to give a systematic computation for rings of invariants for all of the finite classical groups in odd characteristic.
Paper Structure (16 sections, 46 theorems, 193 equations)

This paper contains 16 sections, 46 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. The Setting
  3. Missing Invariants
  4. Steenrod Operations
  5. New Invariants
  6. Formulating the Main Theorem
  7. Sylow Invariants
  8. Computing the Hook Invariants
  9. Computing the Sylow Invariants
  10. Computing the Borel Invariants
  11. The Invariants of
  12. Proof of Lemma \ref{['oplus_tech-lem']}
  13. Proof of Parts (a), (b) and (c)
  14. Proof of Part (d)
  15. Proof of Part (f)
  16. ...and 1 more sections

Key Result

Theorem 2.1

$\mathcal{F}(S_m^{G_m})=\mathbb{F}_q(\xi_1,\ldots,\xi_{n})$.

Theorems & Definitions (98)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • Corollary 3.6
  • Lemma 3.7
  • proof
  • ...and 88 more