Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hurwitz numbers for reflection groups $G(m,1,n)$

Raphaël Fesler, Denis Gorodkov, Maksim Karev

Abstract

We are extending results from \cite{B-Hurwitz} by building a parallel theory of simple Hurwitz numbers for the reflection groups $G(m,1,n)$. We also study analogs of the cut-and-join operators. An algebraic description as well as a description in terms of ramified covering of Hurwitz numbers is provided. An explicit formula for them in terms of Schur polynomials are provided. In addition the generating function of $G(m,1,n)$-Hurwitz numbers is shown to give rise to $m$ independent variables $τ$-function of the KP hierarchy. Finally we provide an ELSV-formula type for these new Hurwitz numbers.

Hurwitz numbers for reflection groups $G(m,1,n)$

Abstract

We are extending results from \cite{B-Hurwitz} by building a parallel theory of simple Hurwitz numbers for the reflection groups . We also study analogs of the cut-and-join operators. An algebraic description as well as a description in terms of ramified covering of Hurwitz numbers is provided. An explicit formula for them in terms of Schur polynomials are provided. In addition the generating function of -Hurwitz numbers is shown to give rise to independent variables -function of the KP hierarchy. Finally we provide an ELSV-formula type for these new Hurwitz numbers.
Paper Structure (15 sections, 20 theorems, 64 equations, 1 figure)

This paper contains 15 sections, 20 theorems, 64 equations, 1 figure.

Table of Contents

  1. Complex reflection groups $G(m,1,n)$
  2. Definitions and embedding into $S_{mn}$
  3. Wreath product, correspondence and conjugacy classes
  4. Hurwitz numbers
  5. Generating function, cut-and-join and explicit formulas
  6. Generating function
  7. Explicit formulas
  8. The operators $\mathcal{CJ}_0$
  9. The operators $\mathcal{CJ}_k$
  10. Change of variables
  11. KP Hierarchy
  12. Hurwitz numbers and ramified covering
  13. Wreath ramified coverings
  14. Hurwitz numbers: the algebro-geometric definition
  15. ELSV-type formula

Key Result

Proposition 1.1

There exist an embedding $\Psi: G(m,1,n) \hookrightarrow S_{mn}$ such that $\Psi(R_{ij}^{(\alpha)})=(i\ \tau^\alpha(j))(\tau(i)\ \tau^{\alpha+1}(j))\dots (\tau^{m-1}(i)\ \tau^{\alpha-1}(j))$ for $\alpha=0,\dots m-1$ and assuming that $\tau^0(j)=j$ and $\Psi(L_i^\alpha)=(i\ \tau(i)\dots \tau^{m-1}

Figures (1)

  • Figure 1: Presentation of the set of $mn$ elements

Theorems & Definitions (35)

  • Proposition 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 1.4
  • Example 1.5
  • proof
  • proof : Proof of proposition \ref{['Prop:Embedding']}
  • Proposition 1.6
  • Definition 2.1
  • Lemma 2.2
  • ...and 25 more