Abstract
A boundary singularity is a singularity of a function on a manifold with boundary. The simple and unimodal boundary singularities were classified by V.I. Arnold and V.I. Matov. The McKay correspondence can be generalized to the simple boundary singularities. We consider the monodromy of the simple, parabolic, and exceptional unimodal boundary singularities. We show that the characteristic polynomial of the monodromy is related to the Poincaré series of the coordinate algebra of the ambient singularity.
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V. I. Arnol’d, “Critical Points of Functions on a Manifold with Boundary, the Simple Lie Groups B k, C k, and F 4 and Singularities of Evolutes,” Usp. Mat. Nauk 33(5), 91–105 (1978) [Russ. Math. Surv. 33 (5), 99–116 (1978)].
V. I. Arnold, V. A. Vasil’ev, V. V. Goryunov, and O. V. Lyashko, Singularities. II: Classification and Applications (VINITI, Moscow, 1989), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 39: Dynamical Systems-8. Engl. transl.: V. I. Arnold, V. V. Goryunov, O. V. Lyashko, and V. A. Vasil’ev, Singularity Theory. II: Classification and Applications (Springer, Berlin, 1993), Encycl. Math. Sci. 39: Dynamical Systems VIII.
V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps: Classification of Critical Points, Caustics and Wave Fronts (Nauka, Moscow, 1982); Engl. transl.: V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhäuser, Boston, 1985), Vol. 1, Monogr. Math. 82.
V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps: Monodromy and Asymptotics of Integrals (Nauka, Moscow, 1984); Engl. transl.: V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhäuser, Boston, 1988), Vol. 2, Monogr. Math. 83.
N. Bourbaki, Éléments de mathématique, Fasc. XXXIV: Groupes et algèbres de Lie, Chapitres 4, 5 et 6 (Hermann, Paris, 1968).
S. Chmutov and I. Scherback, “Dynkin Diagrams of Dual Boundary Singularities Are Dual,” J. Algebr. Geom. 3, 449–462 (1994).
I. V. Dolgachev, “Quotient-Conical Singularities on Complex Surfaces,” Funkts. Anal. Prilozh. 8(2), 75–76 (1974) [Funct. Anal. Appl. 8, 160–161 (1974)].
W. Ebeling, “Milnor Lattices and Geometric Bases of Some Special Singularities,” in Noeuds, tresses et singularit és, Ed. by C. Weber (Enseign. Math. Univ. Genève, Genève, 1983), Monogr. Enseign. Math. 31, pp. 129–146; Enseign. Math. 29, 263–280 (1983).
W. Ebeling, “Poincaré Series and Monodromy of a Two-Dimensional Quasihomogeneous Hypersurface Singularity,” Manuscr. Math. 107, 271–282 (2002).
W. Ebeling, “The Poincaré Series of Some Special Quasihomogeneous Surface Singularities,” Publ. Res. Inst. Math. Sci., Kyoto Univ. 39, 393–413 (2003).
W. Ebeling and D. Ploog, “McKay Correspondence for the Poincaré Series of Kleinian and Fuchsian Singularities,” arXiv: 0809.2738; Math. Ann., doi: 10.1007/s00208-009-0451-4 (2009).
A. M. Gabrielov, “Dynkin Diagrams for Unimodal Singularities,” Funkts. Anal. Prilozh. 8(3), 1–6 (1974) [Funct. Anal. Appl. 8, 192–196 (1974)].
H. Lenzing and J. A. de la Peña, “Extended Canonical Algebras and Fuchsian Singularities,” arXiv:math/0611532.
V. I. Matov, “Unimodal Germs of Functions on a Manifold with a Boundary,” Funkts. Anal. Prilozh. 14(1), 69–70 (1980) [Funct. Anal. Appl. 14, 55–57 (1980)].
J. McKay, “Graphs, Singularities, and Finite Groups,” in The Santa Cruz Conference on Finite Groups (Am. Math. Soc., Providence, RI, 1980), Proc. Symp. Pure Math. 37, pp. 183–186.
P. Slodowy, Simple Singularities and Simple Algebraic Groups (Springer, Berlin, 1980), Lect. Notes Math. 815.
P. Slodowy, “Sur les groupes finis attachés aux singularités simples,” in Introduction à la théorie des singularit és. II (Hermann, Paris, 1988), Trav. Cours 37, pp. 109–126.
R. Stekolshchik, “Kostant’s Generating Functions, Ebeling’s Theorem and McKay’s Observation Relating the Poincaré Series,” arXiv:math/0608500.
R. Stekolshchik, Notes on Coxeter Transformations and the McKay Correspondence (Springer, Berlin, 2008).
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Ebeling, W. Poincaré series and monodromy of the simple and unimodal boundary singularities. Proc. Steklov Inst. Math. 267, 50–58 (2009). https://doi.org/10.1134/S008154380904004X
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DOI: https://doi.org/10.1134/S008154380904004X