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Poincaré series of Klein groups, Coxeter polynomials, the Burau representation, and Milnor invariants. (English. Russian original) Zbl 1251.20010
Proc. Steklov Inst. Math. 267, 139-155 (2009); translation from Tr. Mat. Inst. Steklova 267, 146-163 (2009).
Summary: We obtain several formulas for the Poincaré series defined by B. Kostant for Klein groups (binary polyhedral groups) and some formulas for Coxeter polynomials (characteristic polynomials of monodromy in the case of singularities). Some of these formulas – the generalized Ebeling formula, the Christoffel-Darboux identity, and the combinatorial formula – are corollaries to the well-known statements on the characteristic polynomial of a graph and are analogous to formulas for orthogonal polynomials. The ratios of Poincaré series and Coxeter polynomials are represented in terms of branched continued fractions, which are $$q$$-analogs of continued fractions that arise in the theory of resolution of singularities and in the Kirby calculus. Other formulas connect the ratios of some Poincaré series and Coxeter polynomials with the Burau representation and Milnor invariants of string links. The results obtained by S. M. Gusein-Zade, F. Delgado, and A. Campillo allow one to consider these facts as statements on the Poincaré series of the rings of functions on the singularities of curves, which suggests the following conjecture: the ratio of the Poincaré series of the rings of functions for close (in the sense of adjacency or position in a series) singularities of curves is determined by the Burau representation or by the Milnor invariants of a string link, which is an intermediate object in the transformation of the knot of one singularity into the knot of the other.

##### MSC:
 20C15 Ordinary representations and characters 05E05 Symmetric functions and generalizations 32S25 Complex surface and hypersurface singularities 14J17 Singularities of surfaces or higher-dimensional varieties 14E16 McKay correspondence 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 20F36 Braid groups; Artin groups
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##### References:
 [1] A. B. Givental’, ”Twisted Picard-Lefschetz Formulas,” Funkts. Anal. Prilozh. 22(1), 12–22 (1988) [Funct. Anal. Appl. 22, 10–18 (1988)]. [2] S. M. Gusein-Zade, ”The Monodromy Groups of Isolated Singularities of Hypersurfaces,” Usp. Mat. Nauk 32(2), 23–65 (1977) [Russ. Math. Surv. 32 (2), 23–69 (1977)]. · Zbl 0379.32013 [3] S. M. Gusein-Zade, F. Delgado, and A. Campillo, ”The Alexander Polynomial of Plane Curve Singularities and Rings of Functions on Curves,” Usp. Mat. Nauk 54(3), 157–158 (1999) [Russ. Math. Surv. 54, 634–635 (1999)]. · Zbl 0976.32017 · doi:10.4213/rm160 [4] G. G. Il’yuta, ”Relations for the Alexander Polynomial,” Usp. Mat. Nauk 63(3), 161–162 (2008) [Russ. Math. Surv. 63, 567–569 (2008)]. · doi:10.4213/rm9181 [5] H. S. M. Coxeter, ”The Product of the Generators of a Finite Group Generated by Reflections,” Duke Math. J. 18, 765–782 (1951). · Zbl 0044.25603 · doi:10.1215/S0012-7094-51-01870-4 [6] W. Ebeling, ”Poincaré Series and Monodromy of a Two-Dimensional Quasihomogeneous Hypersurface Singularity,” Manuscr. Math. 107(3), 271–282 (2002). · Zbl 1036.14017 · doi:10.1007/s002290100242 [7] W. Gibson and M. Ishikawa, ”Links and Gordian Numbers Associated with Generic Immersions of Intervals,” Topol. Appl. 123, 609–636 (2002). · Zbl 1028.57006 · doi:10.1016/S0166-8641(01)00224-3 [8] C. D. Godsil, Algebraic Combinatorics (Chapman and Hall, New York, 1993). [9] M. Hirasawa, ”Visualization of A’Campo’s Fibered Links and Unknotting Operation,” Topol. Appl. 121, 287–304 (2002). · Zbl 1016.57006 · doi:10.1016/S0166-8641(01)00124-9 [10] E. Hironaka, ”Chord Diagrams and Coxeter Links,” J. London Math. Soc., Ser. 2, 69, 243–257 (2004). · Zbl 1080.57009 · doi:10.1112/S0024610703004976 [11] O. Holtz and B. Sturmfels, ”Hyperdeterminantal Relations among Symmetric Principal Minors,” J. Algebra 316(2), 634–648 (2007). · Zbl 1130.15005 · doi:10.1016/j.jalgebra.2007.01.039 [12] P. Kirk, C. Livingston, and Z. Wang, ”The Gassner Representation for String Links,” Commun. Contemp. Math. 3(1), 87–136 (2001). · Zbl 0989.57005 · doi:10.1142/S0219199701000299 [13] B. Kostant, ”The McKay Correspondence, the Coxeter Element and Representation Theory,” in The Mathematical Heritage of élie Cartan, Lyon, 1984 (Soc. Math. France, Paris, 1985), Astérisque, Numéro Hors Sér., pp. 209–255. [14] B. Kostant, ”The Coxeter Element and the Branching Law for the Finite Subgroups of SU(2),” in The Coxeter Legacy: Reflections and Projections (Am. Math. Soc., Providence, RI, 2006), pp. 63–70. · Zbl 1139.22008 [15] A. Lascoux and P. Pragacz, ”Bezoutians, Euclidean Algorithm, and Orthogonal Polynomials,” Ann. Comb. 9(3), 301–319 (2005). · Zbl 1073.05070 · doi:10.1007/s00026-005-0259-1 [16] J. Levine, ”A Factorization of the Conway Polynomial,” Comment. Math. Helv. 74, 27–52 (1999). · Zbl 0918.57001 · doi:10.1007/s000140050075 [17] S. Ocken, ”Homology of Branched Cyclic Covers of Knots,” Proc. Am. Math. Soc. 110(4), 1063–1067 (1990). · Zbl 0709.57002 · doi:10.1090/S0002-9939-1990-0984809-5 [18] H. Rosengren, ”Multivariable Christoffel-Darboux Kernels and Characteristic Polynomials of Random Hermitian Matrices,” SIGMA, Symmetry Integrability Geom. Methods Appl. 2, 085 (2006). · Zbl 1132.15008 [19] W. Rossmann, ”McKay’s Correspondence and Characters of Finite Subgroups of SU(2),” in Noncommutative Harmonic Analysis (Birkhäuser, Boston, 2004), Prog. Math. 220, pp. 441–458. · Zbl 1070.20052 [20] H. Seifert, ”Über das Geschlecht von Knoten,” Math. Ann. 110, 571–592 (1935). · Zbl 0010.13303 · doi:10.1007/BF01448044 [21] D. Siersma, ”The Monodromy of a Series of Hypersurface Singularities,” Comment. Math. Helv. 65, 181–197 (1990). · Zbl 0723.32015 · doi:10.1007/BF02566602 [22] R. Stekolshchik, Notes on Coxeter Transformations and the McKay Correspondence (Springer, Berlin, 2008), Springer Monogr. Math. · Zbl 1202.20045 [23] R. Suter, ”Quantum Affine Cartan Matrices, Poincaré Series of Binary Polyhedral Groups, and Reflection Representations,” Manuscr. Math. 122(1), 1–21 (2007). · Zbl 1151.20007 · doi:10.1007/s00229-006-0055-1 [24] T. Tsukamoto and A. Yasuhara, ”A Factorization of the Conway Polynomial and Covering Linkage Invariants,” J. Knot Theory Ramifications 16, 631–640 (2007). · Zbl 1119.57005 · doi:10.1142/S0218216507005403
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