Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory

TL;DR

This paper introduces a comprehensive set of cut-and-join operators in Hurwitz theory, describing their algebraic properties, eigenfunctions, and representations, which unify and generalize previous approaches to Hurwitz numbers.

Contribution

It defines and analyzes the complete set of cut-and-join operators in Hurwitz theory, revealing their algebraic structure and connections to symmetric functions and matrix models.

Findings

Operators have GL characters as eigenfunctions and symmetric-group characters as eigenvalues.

They form a commutative associative algebra called the Universal Hurwitz Algebra.

This algebra encodes Hurwitz numbers as linear forms on Young diagrams.

Abstract

We define cut-and-join operator in Hurwitz theory for merging of two branching points of arbitrary type. These operators have two alternative descriptions:(i) they have the GL characters as eigenfunctions and the symmetric-group characters as eigenvalues; (ii) they can be represented as differential operators of the $W$-type (in particular, acting on the time-variables in the Hurwitz-Kontsevich tau-function). The operators have the simplest form if expressed in terms of the matrix Miwa-variables. They form an important commutative associative algebra, a Universal Hurwitz Algebra, generalizing all group algebra centers of particular symmetric groups which are used in description of the Universal Hurwitz numbers of particular orders. This algebra expresses arbitrary Hurwitz numbers as values of a distinguished linear form on the linear space of Young diagrams, evaluated at the product of all diagrams, which characterize particular ramification points of the covering.