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Polynomial invariants of classical subgroups of $\operatorname{GL}_{2}$: Conjugation over finite fields

Aryaman Maithani

TL;DR

This work analyzes the polynomial invariants of the conjugation action of GL$_2(K)$ and its classical subgroups on spaces of $2\times2$ matrices over a field $K$, with a detailed focus on finite fields. For infinite $K$, invariants are generated by trace and determinant; for finite $K$, the invariant rings become hypersurfaces described via a Noether normalization $R=K[f_1,f_2,f_3,f_4]$ and a single secondary invariant, typically the Jacobian, yielding precise Hilbert series. The paper extends the analysis to traceless, orthogonal, and symmetric subspaces, showing that in each case the invariant ring is either a polynomial ring or a hypersurface, and it characterizes homological properties (Cohen–Macaulay, $a$-invariant), $F$-regularity, and (when applicable) the class group. Key techniques include the construction of primary invariants, use of Steinrod operations over finite fields, and transfers between ambient and subspace representations, with explicit results depending on the parity of the characteristic and the size of the finite field.

Abstract

Consider the conjugation action of the general linear group $\operatorname{GL}_{2}(K)$ on the polynomial ring $K[X_{2 \times 2}]$. When $K$ is an infinite field, the ring of invariants is a polynomial ring generated by the trace and the determinant. We describe the ring of invariants when $K$ is a finite field, and show that it is a hypersurface. We also consider the other classical subgroups, and the polynomial rings corresponding to other subspaces of matrices such as the traceless and symmetric matrices. In each case, we show that the invariant ring is either a polynomial ring or a hypersurface.

Polynomial invariants of classical subgroups of $\operatorname{GL}_{2}$: Conjugation over finite fields

TL;DR

This work analyzes the polynomial invariants of the conjugation action of GL and its classical subgroups on spaces of matrices over a field , with a detailed focus on finite fields. For infinite , invariants are generated by trace and determinant; for finite , the invariant rings become hypersurfaces described via a Noether normalization and a single secondary invariant, typically the Jacobian, yielding precise Hilbert series. The paper extends the analysis to traceless, orthogonal, and symmetric subspaces, showing that in each case the invariant ring is either a polynomial ring or a hypersurface, and it characterizes homological properties (Cohen–Macaulay, -invariant), -regularity, and (when applicable) the class group. Key techniques include the construction of primary invariants, use of Steinrod operations over finite fields, and transfers between ambient and subspace representations, with explicit results depending on the parity of the characteristic and the size of the finite field.

Abstract

Consider the conjugation action of the general linear group on the polynomial ring . When is an infinite field, the ring of invariants is a polynomial ring generated by the trace and the determinant. We describe the ring of invariants when is a finite field, and show that it is a hypersurface. We also consider the other classical subgroups, and the polynomial rings corresponding to other subspaces of matrices such as the traceless and symmetric matrices. In each case, we show that the invariant ring is either a polynomial ring or a hypersurface.
Paper Structure (14 sections, 33 theorems, 64 equations, 1 table)

This paper contains 14 sections, 33 theorems, 64 equations, 1 table.

Table of Contents

  1. Introduction
  2. The action of GL2 on gl2
  3. Preliminaries about the conjugation action
  4. The primary invariants
  5. Homological properties
  6. The missing invariant
  7. Additional properties of the invariant ring
  8. The special linear group and traceless matrices
  9. The action of GL2 on sl2
  10. The action of SL2 on gl2 and sl2
  11. The actions of O2
  12. The action of O2 on gl(2)
  13. The action of O2 on o(2) and symmetric matrices
  14. Steenrod operations

Key Result

Theorem 1.1

Let $K$ be a finite field with $q$ elements. Consider the conjugation action of the general linear group $G \coloneqq \mathop{\mathrm{GL}}\nolimits_{2}(K)$ on the polynomial ring $S \coloneqq K[X_{2 \times 2}]$. Let $f_{1} = \operatorname{trace}$, $f_{2} = \det$, $f_{3} = \mathcal{P}^{1}(\det)$, and Additionally, the invariant ring $S^{G}$ does not split from $S$ (equivalently, $S^{G}$ is not $F$-

Theorems & Definitions (71)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 61 more