Invariants of permutational unitary equivalence classes of the Parseval frames

The unitary equivalence up to a permutation of vectors on the set of frames of a finite-dimensional space is considered. Functions constant on the permutational unitary equivalence classes of the Parseval frames in $\mathbb{C}^n$ are studied. Namely, a set of invariants is given that separates these equivalence classes in general position. Having obtained this result, an algorithm is described that enables one to recover a Parseval frame up to permutational unitary equivalence from the values of the invariants. In this case, the classical questions on the equivalence of rigid frames are considered from an algebraic-geometric point of view. In addition, when proving the main result, the algebraically independent generators of the field of invariants for the action of the symmetric group on the space of selfadjoint matrices are found.