Abstract
There are many papers on the classification of singularities that are invariant or equivariant under the action of a finite group. However, since the problem is difficult, most of these papers consider only special cases, for example, the case of the action of a particular group of small order. In this paper, an attempt is made to prove general statements about equivariantly simple singularities; namely, singularities equivariantly simple with respect to irreducible actions of finite groups are classified. A criterion for the existence of such equivariantly simple singularities is also given.
This is a preview of subscription content, log in via an institution to check access.
References
V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Mappings. Vol. 1: Classification of Critical Points, Caustics, and Wave Fronts, Nauka, Moscow, 1982 (Russian).
E. A. Astashov, “On the classification of singularities that are equivariant simple with respect to representations of cyclic groups”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 26:2 (2016), 155–159.
E. A. Astashov, “Classification of \(\mathbb{Z}_3\)-Equivariant Simple Function Germs”, Mat. Zametki, 105:2 (2019), 163–178; English transl.:, Math. Notes, 105:2 (2019), 161–172.
S. Bochner, “Compact groups of differentiable transformations”, Ann. of Math., 46:3 (1945), 372–381.
S. M. Gusein-Zade and A.-M. Ya. Rauch, “On Simple \({\mathbb Z}_3\)-Invariant Function Germs”, Funktsional. Anal. i Prilozhen., 55:1 (2021), 56–64; , Funct. Anal. Appl., 55:1 (2021), 45–51.
S. M. Gusein-Zade and A.-M. Ya. Raukh, “On Simple \(\mathbb{Z}_2\)-Invariant and Corner Function Germs”, Mat. Zametki, 107:6 (2020), 855–864; English transl.:, Math. Notes, 107:6 (2020), 939–945.
W. Domitrz, M. Manoel, and P. de M. Rios, “The Wigner caustic on shell and singularities of odd functions”, J. Geom. Phys., 71 (2013), 58–72.
I. A. Proskurnin, “Normal forms of equivariant functions”, Vestn. Moskov. Univ. Ser. 1. Mat. Mekh., :2 (2020), 51–55; English transl.:, Moscow Univ. Math. Bull., 75:2 (2020), 83–86.
Funding
This work was supported by the Russian Science Foundation under grant no. 21-11-00080, https://rscf.ru/project/21-11-00080/.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2023, Vol. 57, pp. 77–82 https://doi.org/10.4213/faa4033.
Translated by O. V. Sipacheva
Rights and permissions
About this article
Cite this article
Proskurnin, I.A. Singularities Equivariantly Simple with Respect to Irreducible Representations. Funct Anal Its Appl 57, 60–64 (2023). https://doi.org/10.1134/S0016266323010057
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0016266323010057