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arXiv · 2609.28301

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 $D_4$ and the tetrahedral rotation group $A_4$ recover $D_4$ exactly and all twelve $A_4$ matrices up to modest precision.

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Bernhard G. Bodmann, Reshma Sabnam. 2026-09-23. Single-Orbit Recovery of Groups via an Optimization Principle. https://arxiv.org/abs/2609.28301

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