Invariants of equivalence classes of tight frames
Abstract
We observe the unitary equivalence up to reordering on the set of tight frames in a finite-dimensional space. We show that Parseval frames in Rn in general position are unitarily equivalent up to permutation if and only if certain permutational unitary invariants are equal. This is proved by giving an algorithm to recover a Parseval frame up to permutational unitary equivalence from a some subset of these invariants.
The main result is proved by using the action of the symmetric group on the space of symmetric matrices. More precisely, we show algebraically independent generators of the field of invariants for this action.
Keywords
tight frames,
unitary equivalence,
unitarily equivalence up to reordering,
orbits,
symmetric group,
field of invariants
Supporting agencies: The work is performed under the development program of Volga Region Mathematical Center (agreement no. 075-02-2023-944).
About the Author
V. V. Sevostyanova
Samara National Research University, Volga Region Mathematical Center
Russian Federation
Russian Federation
Victoria Vladimirovna Sevostyanova
1 Akad. Pavlova str., Samara, 443011
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Review
For citations:
Sevostyanova V.V. Invariants of equivalence classes of tight frames. Mathematics and Theoretical Computer Science. 2023;1(3):46-58. (In Russ.)
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