One generic complex orbit determines a finite linear symmetry group up to isomorphism; two generic real orbits suffice, and concrete recovery needs an orbit count governed by representation multiplicities.
Radon-Hurwitz Grassmannian codes
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abstract
Every equi-isoclinic tight fusion frame (EITFF) is a type of optimal code in a Grassmannian, consisting of subspaces of a finite-dimensional Hilbert space for which the smallest principal angle between any pair of them is as large as possible. EITFFs yield dictionaries with minimal block coherence and so are ideal for certain types of compressed sensing. By refining classical work of Lemmens and Seidel based on Radon-Hurwitz theory, we fully characterize EITFFs in the special case where the dimension of the subspaces is exactly one-half of that of the ambient space. We moreover show that each such "Radon-Hurwitz EITFF" is highly symmetric, where every even permutation is an automorphism.
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Recovering a group from few orbits
One generic complex orbit determines a finite linear symmetry group up to isomorphism; two generic real orbits suffice, and concrete recovery needs an orbit count governed by representation multiplicities.