The two-step spectral locus of the octonionic structure operator
Abstract
We continue the study of the octonionic spectrum of the octonionic structure operator h on W=𝕆⊗ℝV, where V is the seven-dimensional representation of G2. The real spectrum and the octonionic one-step spectrum, given by the equation h(w)=wλ for real or octonion λ, have been determined in earlier work. In this paper, we study the two-step equation h(w)=(wλ1)λ2. Due to non-associativity, this does not in general reduce to the one-step problem. An adapted quaternionic frame represents successive right multiplication by an operator with four real coefficients. The SU(2) subgroup fixing the frame splits W into invariant sectors of dimensions $16$, $12$ and $28$. We factor the determinant of the resulting 4-parameter operator pencil and characterize the pairs for which a nonzero solution exists. The singular locus of this pencil in ℝ4 is the union of an explicit algebraic set and a round $3$-sphere. The $12$-dimensional sector carries the one-step problem at the reversed product λ2λ1. The $28$-dimensional sector contributes no additional real singular points; the fixed sector produces solutions even when neither product lies in the one-step right spectrum.
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BibTeX
@misc{grigorian2026twostep,
title = {{The two-step spectral locus of the octonionic structure operator}},
author = {Sergey Grigorian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26613},
eprint = {2610.26613},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26613},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The two-step spectral locus of the octonionic structure operator
AU - Sergey Grigorian
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26613
UR - https://arxiv.org/abs/2610.26613
AB - We continue the study of the octonionic spectrum of the octonionic structure operator h on W=𝕆⊗_ℝV, where V is the seven-dimensional representation of G_2. The real spectrum and the octonionic one-step spectrum, given by the equation h(w)=wλ for real or octonion λ, have been determined in earlier work. In this paper, we study the two-step equation h(w)=(wλ_1)λ_2. Due to non-associativity, this does not in general reduce to the one-step problem. An adapted quaternionic frame represents successive right multiplication by an operator with four real coefficients. The SU(2) subgroup fixing the frame splits W into invariant sectors of dimensions $16$, $12$ and $28$. We factor the determinant of the resulting 4-parameter operator pencil and characterize the pairs for which a nonzero solution exists. The singular locus of this pencil in ℝ^4 is the union of an explicit algebraic set and a round $3$-sphere. The $12$-dimensional sector carries the one-step problem at the reversed product λ_2λ_1. The $28$-dimensional sector contributes no additional real singular points; the fixed sector produces solutions even when neither product lies in the one-step right spectrum.
ER -