Geodesic
On Simple ${\mathbb Z}_3$-Invariant Function Germs
Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 1, pp. 56-64.

Voir la notice de l'article provenant de la source Math-Net.Ru

V. I. Arnold classified simple (i.e., having no moduli for classification) singularities (function germs) and also simple boundary singularities, that is, function germs invariant with respect to the action $\sigma(x_1; y_1,\dots, y_n)=(-x_1; y_1,\dots, y_n)$ of the group ${\mathbb Z}_2$. In particular, he showed that a function germ (a germ of a boundary singularity) is simple if and only if the intersection form (respectively, the restriction of the intersection form to the subspace of anti-invariant cycles) of a germ in $3+4s$ variables stably equivalent to the one under consideration is negative definite and if and only if the (equivariant) monodromy group on the corresponding space is finite. In a previous paper the authors obtained analogues of the latter statements for function germs invariant with respect to an arbitrary action of the group ${\mathbb Z}_2$ and also for corner singularities. This paper presents an analogue of the simplicity criterion in terms of the intersection form for functions invariant with respect to a number of actions (representations) of the group ${\mathbb Z}_3$.
Mots-clés : Group action, invariant germ
Keywords: simple singularity. 
@article{FAA_2021_55_1_a4,
     author = {S. M. Gusein-Zade and A.-M. Ya. Rauch},
     title = {On {Simple} ${\mathbb Z}_3${-Invariant} {Function} {Germs}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {56--64},
     publisher = {mathdoc},
     volume = {55},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2021_55_1_a4/}
}
TY  - JOUR
AU  - S. M. Gusein-Zade
AU  - A.-M. Ya. Rauch
TI  - On Simple ${\mathbb Z}_3$-Invariant Function Germs
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2021
SP  - 56
EP  - 64
VL  - 55
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2021_55_1_a4/
LA  - ru
ID  - FAA_2021_55_1_a4
ER  - 
%0 Journal Article
%A S. M. Gusein-Zade
%A A.-M. Ya. Rauch
%T On Simple ${\mathbb Z}_3$-Invariant Function Germs
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2021
%P 56-64
%V 55
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2021_55_1_a4/
%G ru
%F FAA_2021_55_1_a4
S. M. Gusein-Zade; A.-M. Ya. Rauch. On Simple ${\mathbb Z}_3$-Invariant Function Germs. Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 1, pp. 56-64. http://geodesic.mathdoc.fr/item/FAA_2021_55_1_a4/

[1] V. I. Arnold, “Normalnye formy funktsii vblizi vyrozhdennykh kriticheskikh tochek, gruppy Veilya $A_k$, $D_k$, $E_k$ i lagranzhevy osobennosti”, Funkts. analiz i ego pril., 6:4 (1972), 3–25 | DOI | MR

[2] V. I. Arnold, “Kriticheskie tochki funktsii na mnogoobrazii s kraem, prostye gruppy Li $B_k$, $C_k$, $F_4$ i osobennosti evolyut”, UMN, 33:5(203) (1978), 91–105 | MR | Zbl

[3] V. I. Arnold, A. N. Varchenko, S. M. Gusein-Zade, Osobennosti differentsiruemykh otobrazhenii, v. 1, Klassifikatsiya kriticheskikh tochek, kaustik i volnovykh frontov, Nauka, M., 1982

[4] M. Kreuzer, “The mirror map for invertible LG models”, Phys. Lett. B, 328:3–4 (1994), 312–318 | DOI | MR

[5] S. M. Gusein-Zade, A.-M. Ya. Raukh, “O prostykh ${\mathbb Z}_2$-invariantnykh i uglovykh rostkakh funktsii”, Matem. zametki, 107:6 (2020), 855–864 | DOI | MR | Zbl

[6] P. Slodowy, “Einige Bemerkungen zur Entfaltung symmetrischer Funktionen”, Math. Z., 158:2 (1978), 157–170 | DOI | MR | Zbl

[7] J. Steenbrink, “Intersection form for quasi-homogeneous singularities”, Compositio Math., 34:2 (1977), 211–223 | MR | Zbl

[8] C. T. C. Wall, “A note on symmetry of singularities”, Bull. London Math. Soc., 12:3 (1980), 169–175 | DOI | MR | Zbl

[9] G. Wassermann, “Classification of singularities with compact abelian symmetry”, Singularities, Warsaw, 1985, Banach Center Publ., 20, PWN, Warsaw, 1988, 475–498 | DOI | MR