Other Articles by Authors

Evgeny Astashov

Authors and WSEAS

Evgeny Astashov

WSEAS Transactions on Mathematics

Print ISSN: 1109-2769
E-ISSN: 2224-2880

Volume 17, 2018

Notice: As of 2014 and for the forthcoming years, the publication frequency/periodicity of WSEAS Journals is adapted to the 'continuously updated' model. What this means is that instead of being separated into issues, new papers will be added on a continuous basis, allowing a more regular flow and shorter publication times. The papers will appear in reverse order, therefore the most recent one will be on top.

Volume 17, 2018

Equivariant Simple Singularities and Admissible Sets of Weights

AUTHORS: Evgeny Astashov

Download as PDF

Equivariant maps, i.e., maps that commute with group actions on the source and target, play an important role in the study of manifolds with group actions. It is therefore of interest to classify equivariant maps up to certain equivalence relations. In this paper we study multivariate holomorphic function germs that are equivariant with respect to finite cyclic groups. The natural equivalence relation between such germs is provided by the action of the group of biholomorphic automorphism germs of the source. An orbit of this action is called equivariant simple if its sufficiently small neighborhood intersects only a finite number of other orbits. We present a sufficient condition under which there exist no singular equivariant holomorphic function germs; it is also shown that this condition is not necessary. The condition is formulated in terms of admissible sets of weights; such sets are defined and classified for all finite cyclic group representations. As an application we describe scalar actions of finite cyclic groups for which there exist no equivariant simple singular function germs.

KEYWORDS: Equivariant maps, finite group actions, singularity theory, classification of singularities, simple singularities.


[1] V. I. Arnold, Normal forms of functions near degenerate critical points, the Weyl groups Ak, Dk, Ek and Lagrangian singularities, Functional Anal. Appl. 6:4, 1972, pp. 254–272.

[2] V. I. Arnold, Indices of singular points of 1- forms on a manifold with boundary, convolution of invariants of reflection groups, and singular projections of smooth surfaces, Russian Math. Surveys 34:1, 1979, pp. 1–42.

[3] V. V. Goryunov, Simple Functions on Space Curves, Funktsional. Anal. Appl. 34:2, 2000, PP. 129–132.

[4] W. Domitrz, M. Manoel, P. de M. Rios. The Wigner caustic on shell and singularities of odd functions, Journal of Geometry and Physics 71, 2013, pp. 58–72.

[5] E. A. Astashov, On the classification of singularities that are equivariant simple with respect to representations of cyclic groups (in Russian), Bulletin of Udmurt University. Mathematics, Mechanics, Computer Science 26:2, 2016, pp. 155-159.

[6] E. A. Astashov. On the classification of bivariate function germs singularities that are equivariant simple with respect to the cyclic group of order three (in Russian), Bulletin of Samara University. Natural sciences 3-4, 2016, pp. 7-13.

[7] J.W. Bruce, On Families of Square Matrices, ˙ Cadernos de matematica ´ 3, 2002, pp. 217-242.

[8] J.W. Bruce, On Families of Symmetric Matri- ˙ ces, Moscow Mathematical Journal 3:2, 2003, pp. 335-360.

[9] V. V. Goryunov, V. M. Zakalyukin, Simple symmetric matrix singularities and the subgroups of Weyl groups Aµ, Dµ, Eµ, Moscow Mathematical Journal 3:2, 2003, pp. 507-530.

[10] C. E. Baines, V. V. Goryunov, Cyclically equivariant function singularities and unitary reflection groups G(2m, 2, n), G9, G31, St-Petersburg Mathematical Journal 11:5, 2000, pp. 761-774.

[11] P. H. Baptistelli, M. G. Manoel, The classification of reversible-equivariant steady-state bifurcations on self-dual spaces, Math. Proc. Cambridge Philos. Soc. 145:2, 2008, pp. 379–401.

[12] M. Manoel, I. O. Zeli, Complete transversals of reversible equivariant singularities of vector fields, arXiv:1309.1904

[math.DS], 2013.

[13] P. H. Baptistelli, M. Manoel, I. O. Zeli, The classification of reversible-equivariant steady-state bifurcations on self-dual spaces, Bull. Braz. Math. Soc., New Series 47:3, 2016, pp. 935–954.

[14] M. Manoel, P. Tempesta, On equivariant binary differential equations, arXiv:1608.05575

[math.DS], 2016.

[15] P. H. Baptistelli, M. Manoel, I. O. Zeli, Normal forms of bireversible vector fields, arXiv:1702.04658

[math.DS], 2017.

[16] P. Slodowy, Einige Bemerkungen zur Entfaltung symmetrischer Funktionen, Math. Z. 158, 1978, 157–170.

[17] J. W. Bruce,A. A. du Plessis, C. T. C. Wall, Determinacy and unipotency, Invent. Math 88, 1987, 521–54.

[18] M. Golubitsky, I. Stewart, D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, vol. II, Springer-Verlag, New York, 1988.

[19] S. Bochner, Compact groups of differentiable transformations, Ann. Math. 2:46, 1945, pp. 372–381.

[20] J. W. Milnor, Morse theory, Princeton University Press, 1963.

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 17, 2018, Art. #49, pp. 404-410

Copyright Β© 2018 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution License 4.0

Bulletin Board


The editorial board is accepting papers.

WSEAS Main Site