|
Section
|
Mathematics
|
|
Title
|
On the classification of singularities that are equivariant simple with respect to representations of cyclic groups
|
|
Author(-s)
|
Astashov E.A.a
|
|
Affiliations
|
Lomonosov Moscow State Universitya
|
|
Abstract
|
We consider the problem of classification of function germs $(\mathbb{C}^n, 0)\to(\mathbb{C}, 0)$ that are equivariant simple with respect to various representations of a finite cyclic group $\mathbb{Z}_m$, $m\geqslant 3$, on $\mathbb{C}^n$ and $\mathbb{C}$ up to equivariant automorphisms of $\mathbb{C}^n$. In the case of matching scalar actions of the group it is shown that for $n\geqslant 2$ there exist no equivariant simple function germs. This result is generalized to the cases where the group action in several variables in $\mathbb{C}^n$ coincides with the action of the group on $\mathbb{C}$. In addition, it is shown that in the case of non-matching scalar actions of $\mathbb{Z}_3$ on $\mathbb{C}^2$ and on $\mathbb{C}$ any equivariant simple function germ is equivalent to one of the germs $A_{3k+1}$, $k\in\mathbb{Z}_{\geqslant 0}$.
|
|
Keywords
|
classification of singularities, simple singularities, group action, equivariant functions
|
|
UDC
|
512.761.5
|
|
MSC
|
14B05
|
|
DOI
|
10.20537/vm160201
|
|
Received
|
12 May 2016
|
|
Language
|
Russian
|
|
Citation
|
Astashov E.A. On the classification of singularities that are equivariant simple with respect to representations of cyclic groups, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 2, pp. 155-159.
|
|
References
|
- Arnol'd V.I. Normal forms of functions near degenerate critical points, the Weyl groups of $A_k$, $D_k$, $E_k$ and Lagrangian singularities, Funct. Anal. Appl., 1972, vol. 6, no. 4, pp. 254-272.
- Arnol'd V.I. Critical points of functions on a manifold with boundary, the simple Lie groups $B_k$, $C_k$, and $F_4$ and singularities of evolutes, Russian Mathematical Surveys, 1978, vol. 33, no. 5, pp. 99-116.
- Domitrz W., Manoel M., Rios P. de M. The Wigner caustic on shell and singularities of odd functions, Journal of Geometry and Physics, 2013, vol. 71, pp. 58-72.
- Bruce J.W., Kirk N.P., du Plessis A.A. Complete transversals and the classification of singularities, Nonlinearity, 1997, vol. 10, pp. 253-275.
- Arnold V.I., Gusein-Zade S.M., Varchenko A.N. Singularities of differentiable maps, Volumes I-II, Boston: Birkhauser, 1985-1988 (Monographs Math., vol. 82-83).
|
|
Full text
|
|
+7 (3412) 91 60 92
Russian