Geodesic
Unitary reflection groups associated with singularities of functions with cyclic symmetry
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 54 (1999) no. 5, pp. 873-893 
V. V. Goryunov. Unitary reflection groups associated with singularities of functions with cyclic symmetry. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 54 (1999) no. 5, pp. 873-893. http://geodesic.mathdoc.fr/item/RM_1999_54_5_a0/
@article{RM_1999_54_5_a0,
     author = {V. V. Goryunov},
     title = {Unitary reflection groups associated with singularities of functions with cyclic symmetry},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {873--893},
     year = {1999},
     volume = {54},
     number = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_1999_54_5_a0/}
}
TY  - JOUR
AU  - V. V. Goryunov
TI  - Unitary reflection groups associated with singularities of functions with cyclic symmetry
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 1999
SP  - 873
EP  - 893
VL  - 54
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/RM_1999_54_5_a0/
LA  - en
ID  - RM_1999_54_5_a0
ER  - 
%0 Journal Article
%A V. V. Goryunov
%T Unitary reflection groups associated with singularities of functions with cyclic symmetry
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 1999
%P 873-893
%V 54
%N 5
%U http://geodesic.mathdoc.fr/item/RM_1999_54_5_a0/
%G en
%F RM_1999_54_5_a0

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

Finite groups generated by Euclidean reflections have been commonplace in various problems of singularity theory since their relationship with the classification of critical points of functions was discovered by Arnol'd [1], [2]. We show that a number of finite groups generated by unitary reflections are also naturally related to singularities of functions, namely, those invariant under a unitary reflection of finite order. To this end, we consider germs of functions on a manifold with boundary and lift them to a cyclic covering of the manifold, ramified over the boundary. This construction provides a new notion of roots for the groups under consideration and provides skew-Hermitian analogues of these groups.

[1] Arnold V. I., “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 | MR | Zbl

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

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

[4] “Monodromiya izolirovannykh osobennostei giperpoverkhnostei”, Matematika, 15:4 (1972), 130–160

[5] Brieskorn E., “Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe”, Invent. Math., 12 (1971), 57–61 | DOI | MR | Zbl

[6] Broué M., Malle G., “Zyklotomische Heckealgebren”, Astérisque, 212, 1993, 119–189 | MR | Zbl

[7] Cohen A. M., “Finite complex reflection groups”, Ann. Sci. Ecole Norm. Sup., 9 (1976), 379–436 | MR | Zbl

[8] Coxeter H. S. M., “Finite groups generated by unitary reflections”, Abh. Math. Sem. Univ. Hamburg, 31 (1967), 125–135 | DOI | MR | Zbl

[9] Coxeter H. S. M., Regular Complex Polytopes, Cambridge Univ. Press, London, 1974 | Zbl

[10] Ebeling W., Gusein-Zade S. M., “Suspensions of fat points and their intersection forms”, Progr. Math., 162, 1998, 141–165 | MR | Zbl

[11] Gabrielov A. M., “Matritsy peresechenii dlya nekotorykh osobennostei”, Funkts. analiz i ego pril., 7:3 (1973), 8–32 | MR

[12] Givental A. B., “Skruchennye formuly Pikara–Lefshetsa”, Funkts. analiz i ego pril., 22:1 (1988), 12–22 | MR | Zbl

[13] Givental A. B., “Osobye lagranzhevy mnogoobraziya i ikh lagranzhevy otobrazheniya”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Noveishie dostizheniya, 33, VINITI, M., 1988, 55–112 | MR

[14] Lyashko O. V., “Klassifikatsiya kriticheskikh tochek funktsii na mnogoobrazii s osobym kraem”, Funkts. analiz i ego pril., 17:3 (1983), 28–36 | MR | Zbl

[15] Milnor Dzh., Osobye tochki kompleksnykh giperpoverkhnostei, Mir, M., 1971 | Zbl

[16] Orlik P., Solomon L., “Discriminants in the invariant theory of reflection groups”, Nagoya Math. J., 109 (1988), 23–45 | MR | Zbl

[17] Popov V. L., “Discrete complex reflection groups”, Comm. Math. Inst., 15, Rijksuniv. Utr., 1982, 1–89

[18] Scherbak O. P., “Volnovye fronty i gruppy otrazhenii”, UMN, 43:3 (1988), 125–160 | MR | Zbl

[19] Shephard G. C., Todd J. A., “Finite unitary reflection groups”, Canad. J. Math., 6 (1954), 274–304 | MR | Zbl

[20] Siersma D., “Singularities of functions on boundaries, corners, etc.”, Quart. J. Math., 32 (1981), 119–127 | DOI | MR | Zbl

[21] Tibar M., “On isolated cyclic quotients with simplest complex link”, Abh. Math. Sem. Univ. Hamburg, 65 (1995), 205–214 | DOI | MR | Zbl

[22] Wassermann G., “Classification of singularities with compact abelian symmetry”, Regensburger Math. Schr., 1 (1977), 1–284 | MR | Zbl