Quantum Hurwitz numbers and Macdonald polynomials

Abstract

Parametric families in the centre $\mathbf{Z}(\mathbf{C}[S_n])$ 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.

23 pages. Minor typos corrected. References updated. Scalar product for Jack polynomials corrected

References in corpus (9)

• On KP-integrable Hurwitz functions
• Hypergeometric τ-functions, Hurwitz numbers and enumeration of paths
• Virasoro constraints and topological recursion for Grothendieck's dessin counting
• The matrix model for dessins d'enfants
• The matrix model for hypergeometric Hurwitz numbers
• Generating functions for weighted Hurwitz numbers
• Reflection factorizations of Singer cycles
• Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers
• Multispecies Weighted Hurwitz Numbers

Cited by in corpus (8)

• Toda hierarchies and their applications
• Weighted Hurwitz numbers and topological recursion: an overview
• Weighted Hurwitz numbers and topological recursion
• Fermionic approach to weighted Hurwitz numbers and topological recursion
• Rationally weighted Hurwitz numbers, Meijer $G$-functions and matrix integrals
• Generating weighted Hurwitz numbers
• Matrix model generating function for quantum weighted Hurwitz numbers
• Asymptotics of quantum weighted Hurwitz numbers