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Classification of G_2-orbits for pairs of octonions

Artem Lopatin, Alexandr N. Zubkov

TL;DR

The paper addresses the problem of classifying ${\rm G}_2$-orbits on pairs of octonions by constructing an explicit minimal transversal for the diagonal action on $\mathbf{O}^2$. It employs the structure of the split Cayley octonion algebra, stabilizer analysis in ${\rm G}_2$, and polynomial invariants, including the trace $\mathrm{tr}$, the norm $n$, and a scalar function $\mathrm{scal}$, to derive complete orbit representatives. The main contributions are explicit representatives for single octonions and for pairs, along with a separating set of invariants that discriminates orbits, and applications to ${\rm G}_2$-invariants in characteristic two. These results enhance understanding of ${\rm G}_2$-subgroup actions and provide practical tools for solving octonionic polynomial equations and for invariants computation in algebraic contexts.

Abstract

Over an algebraically closed field, we described a minimal set of representatives for G_2-orbits on the set of pairs of octonions.

Classification of G_2-orbits for pairs of octonions

TL;DR

The paper addresses the problem of classifying -orbits on pairs of octonions by constructing an explicit minimal transversal for the diagonal action on . It employs the structure of the split Cayley octonion algebra, stabilizer analysis in , and polynomial invariants, including the trace , the norm , and a scalar function , to derive complete orbit representatives. The main contributions are explicit representatives for single octonions and for pairs, along with a separating set of invariants that discriminates orbits, and applications to -invariants in characteristic two. These results enhance understanding of -subgroup actions and provide practical tools for solving octonionic polynomial equations and for invariants computation in algebraic contexts.

Abstract

Over an algebraically closed field, we described a minimal set of representatives for G_2-orbits on the set of pairs of octonions.
Paper Structure (11 sections, 8 theorems, 34 equations)

This paper contains 11 sections, 8 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Octonions
  4. The group ${\rm G}_2$
  5. Notations
  6. Auxiliaries
  7. Polynomial ${\rm G}_2$-invariants
  8. Sets of octonions
  9. Stabilizers in ${\rm G}_2$
  10. Classification of octonions
  11. Classification of a pair of octonions

Key Result

Lemma 2.1

Given $\alpha,\beta\in \mathbb{F}$ with $\alpha\neq\beta$, the stabilizer $\mathop{\rm St}_{{\rm G}_2}(\alpha e_1 + \beta e_2)$ is equal to ${\rm SL}_3$.

Theorems & Definitions (16)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 4.1
  • ...and 6 more