Geodesic
Invariants of permutational unitary equivalence classes of the Parseval frames
Matematičeskie zametki, Tome 116 (2024) no. 4, pp. 584-598 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: rigid frame, Parseval frame, unitary equivalence, permutational unitary equivalence, symmetric group, field of invariants.
Mots-clés : orbit 
@article{MZM_2024_116_4_a7,
     author = {V. V. Sevost'yanova},
     title = {Invariants of permutational unitary equivalence classes of the {Parseval} frames},
     journal = {Matemati\v{c}eskie zametki},
     pages = {584--598},
     year = {2024},
     volume = {116},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2024_116_4_a7/}
}
TY  - JOUR
AU  - V. V. Sevost'yanova
TI  - Invariants of permutational unitary equivalence classes of the Parseval frames
JO  - Matematičeskie zametki
PY  - 2024
SP  - 584
EP  - 598
VL  - 116
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/MZM_2024_116_4_a7/
LA  - ru
ID  - MZM_2024_116_4_a7
ER  - 
%0 Journal Article
%A V. V. Sevost'yanova
%T Invariants of permutational unitary equivalence classes of the Parseval frames
%J Matematičeskie zametki
%D 2024
%P 584-598
%V 116
%N 4
%U http://geodesic.mathdoc.fr/item/MZM_2024_116_4_a7/
%G ru
%F MZM_2024_116_4_a7
V. V. Sevost'yanova. Invariants of permutational unitary equivalence classes of the Parseval frames. Matematičeskie zametki, Tome 116 (2024) no. 4, pp. 584-598. http://geodesic.mathdoc.fr/item/MZM_2024_116_4_a7/

[1] O. Christensen, An Introduction to Frames and Riesz Bases, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, MA, 2002 | MR

[2] S. F. D. Waldron, An Introduction to Finite Tight Frames, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, MA, 2018 | MR

[3] M. Fickus, J. Jasper, E. J. King, D. G. Mixon, “Equiangular tight frames that contain regular simplices”, Linear Algebra Appl., 555 (2018), 98–138 | DOI | MR

[4] S. Ya. Novikov, “Equiangular tight frames with simplices and with full spark in $\mathbb{R}^d$”, Lobachevskii J. Math., 42:1 (2021), 155–166 | DOI | MR

[5] M. N. Istomina, A. B. Pevnyi, “O raspolozhenii tochek na sfere i freime Mersedes–Bents”, Matem. prosv., ser. 3, 11, Izd-vo MTsNMO, M., 2007, 105–112

[6] A. Abdollahi, H. Najafi, “Frame graph”, Linear Multilinear Algebra, 66:6 (2018), 1229–1243 | DOI | MR

[7] V. V. Sevostyanova, “Invarianty na klassakh ekvivalentnosti zhestkikh freimov”, Matem. teoret. komp. nauki, 1:3 (2023), 46–58

[8] S. Ya. Novikov, V. V. Sevostyanova, “Klassy ekvivalentnosti freimov Parsevalya”, Matem. zametki, 112:6 (2022), 850–866 | DOI | MR

[9] Dzh. Kharris, Algebraicheskaya geometriya. Nachalnyi kurs, MTsNMO, M., 2005 | MR

[10] Kh. Kraft, Geometricheskie metody v teorii invariantov, Mir, M., 1987 | MR

[11] E. B. Vinberg, V. L. Popov, “Teoriya invariantov”, Algebraicheskaya geometriya – 4, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 55, VINITI, M., 1989, 137–309 | MR | Zbl

[12] S. Ya. Novikov, V. V. Sevostyanova, “Ravnomernye zhestkie freimy Maltseva”, Izv. RAN. Ser. matem., 86:4 (2022), 162–174 | DOI | MR