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PreprintarXiv licencearXiv

Single-Orbit Recovery of Groups via an Optimization Principle

  • Bernhard G. Bodmann
  • Reshma Sabnam
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Abstract

The objective of this paper is to identify a finite group based on the orbit of a vector under its action. When a finite group acts unitarily on a Hilbert space, the associated Gram matrix lies in the group algebra and satisfies identities that encode the algebraic group structure. We ask about the converse: Given a Gram matrix obtained from the orbit of a vector under the group action, can one recover the underlying \emph{abstract} group structure? Our main result identifies a short list of conditions on a family of orthogonal matrices that force the family to be the right regular representation of a finite group. Building on this characterization, we develop a staged optimization framework that enforces these conditions progressively. Numerical experiments on the dihedral group D4 and the tetrahedral rotation group A4 recover D4 exactly and all twelve A4 matrices up to modest precision.

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Cite

BibTeXRIS
BibTeX
@misc{bodmann2026singleorbit,
  title = {{Single-Orbit Recovery of Groups via an Optimization Principle}},
  author = {Bernhard G. Bodmann and Reshma Sabnam},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28301},
  eprint = {2609.28301},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28301},
}
RIS
TY  - UNPB
TI  - Single-Orbit Recovery of Groups via an Optimization Principle
AU  - Bernhard G. Bodmann
AU  - Reshma Sabnam
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28301
UR  - https://arxiv.org/abs/2609.28301
AB  - The objective of this paper is to identify a finite group based on the orbit of a vector under its action. When a finite group acts unitarily on a Hilbert space, the associated Gram matrix lies in the group algebra and satisfies identities that encode the algebraic group structure. We ask about the converse: Given a Gram matrix obtained from the orbit of a vector under the group action, can one recover the underlying \emph{abstract} group structure? Our main result identifies a short list of conditions on a family of orthogonal matrices that force the family to be the right regular representation of a finite group. Building on this characterization, we develop a staged optimization framework that enforces these conditions progressively. Numerical experiments on the dihedral group D_4 and the tetrahedral rotation group A_4 recover D_4 exactly and all twelve A_4 matrices up to modest precision.
ER  -