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

E. A. Astashov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 26 (2016) no. 2, pp. 155-159

Résumé

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 $A_{3k+1}$, $k\in\mathbb{Z}_{\ge0}$.

Détail

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