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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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