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Wreath Hurwitz numbers, colored cut-and-join equations, and 2-Toda hierarchy

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Abstract

Let G be arbitrary finite group, define H • G (t; p +, p −) to be the generating function of G-wreath double Hurwitz numbers. We prove that H • G (t; p +, p −) satisfies a differential equation called the colored cut-and-join equation. Furthermore, H • G (t; p +, p −) is a product of several copies of tau functions of the 2-Toda hierarchy, in independent variables. These generalize the corresponding results for ordinary Hurwitz numbers.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    HanXiong Zhang & Jian Zhou

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  1. HanXiong Zhang
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  2. Jian Zhou
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Correspondence to Jian Zhou.

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Zhang, H., Zhou, J. Wreath Hurwitz numbers, colored cut-and-join equations, and 2-Toda hierarchy. Sci. China Math. 55, 1627–1646 (2012). https://doi.org/10.1007/s11425-012-4383-1

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  • DOI: https://doi.org/10.1007/s11425-012-4383-1

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