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Abstract

This is an overview of recent results on the use of 2D Toda $\tau$-functions
as generating functions for multiparametric families of weighted Hurwitz
numbers. The Bose-Fermi equivalence composed with the characteristic map
provides an isomorphism between the zero charge sector of the Fermionic Fock
space and the direct sum of the centers of the group algebra of the symmetric
groups $S_n$. Specializing the fermionic formula to the case of diagonal group
elements gives $\tau$-functions of hypergeometric type, for which the expansion
over products of Schur functions is diagonal, with coefficients of {\em content
product} type. The corresponding abelian group action on the centre of the
$S_n$ group algebra is determined by forming symmetric functions
multiplicatively from a weight generating function $G(z)$ and evaluating on the
Jucys-Murphy elements of the group algebra. The resulting central elements act
diagonally on the basis of orthogonal idempotents and the eigenvalues
$r^{G(z)}_\lambda$ are the {\em content product} coefficients appearing in the
double Schur function expansion. Both the geometrical meaning of weighted
Hurwitz numbers, as weighted sums over $n$-sheeted branched coverings, and the
combinatorial one, as weighted enumeration of paths in the Cayley graph of
$S_n$ generated by transpositions follow from expansion of the
Cauchy-Littlewood generating functions over dual pairs of bases of the algebra
of symmetric functions. The coefficients in the resulting $\tau$-function
expansion over products of power sum symmetric functions are the weighted
Hurwitz numbers. Replacement of the Cauchy-Littlewood generating function by
that for Macdonald polynomials provides $(q,t)$-deformations that yield
generating functions for quantum weighted Hurwitz numbers.