Quantum Hurwitz numbers and Macdonald polynomials

TL;DR

This paper introduces quantum Hurwitz numbers derived from Macdonald polynomials, linking algebraic, geometric, and combinatorial perspectives to enumerate weighted branched coverings and paths in symmetric groups.

Contribution

It establishes a novel connection between Macdonald polynomials, quantum Hurwitz numbers, and enumerative geometry, providing a unified framework for weighted coverings and symmetric group paths.

Findings

Eigenvalues as coefficients in 2D Toda $ au$-functions

Interpretation of coefficients as quantum weighted sums

Connection between algebraic, geometric, and combinatorial structures

Abstract

Parametric families in the centre ${\bf Z}({\bf C}[S_n])$ of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements. Their eigenvalues provide coefficients in the double Schur function expansion of 2D Toda $\tau$-functions of hypergeometric type. Expressing these in the basis of products of power sum symmetric functions, the coefficients may be interpreted geometrically as parametric families of quantum Hurwitz numbers, enumerating weighted branched coverings of the Riemann sphere. Combinatorially, they give quantum weighted sums over paths in the Cayley graph of $S_n$ generated by transpositions. Dual pairs of bases for the algebra of symmetric functions with respect to the scalar product in which the Macdonald polynomials are orthogonal provide both the geometrical and combinatorial significance of these quantum weighted enumerative invariants.