This article is cited in 6 papers

MATHEMATICS

On the classification of singularities that are equivariant simple with respect to representations of cyclic groups

E. A. Astashov

Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskie Gory, 1, GSP-1, Moscow, 119991, Russia

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 $\mathbb{Z}_m$, $m\ge3$, 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 $n\ge2$ 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 $A_{3k+1}$, $k\in\mathbb{Z}_{\ge0}$.

Keywords: classification of singularities, simple singularities, group action, equivariant functions.

UDC: 512.761.5

MSC: 14B05

Received: 12.05.2016

DOI: 10.20537/vm160201

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