  We derive the analog of the large $N$ Gross-Taylor holomorphic string
expansion for the refinement of $q$-deformed $U(N)$ Yang-Mills theory on a
compact oriented Riemann surface. The derivation combines Schur-Weyl duality
for quantum groups with the Etingof-Kirillov theory of generalized quantum
characters which are related to Macdonald polynomials. In the unrefined limit
we reproduce the chiral expansion of $q$-deformed Yang-Mills theory derived by
de Haro, Ramgoolam and Torrielli. In the classical limit $q=1$, the expansion
defines a new $\beta$-deformation of Hurwitz theory wherein the refined
partition function is a generating function for certain parameterized Euler
characters, which reduce in the unrefined limit $\beta=1$ to the orbifold Euler
characteristics of Hurwitz spaces of holomorphic maps. We discuss the
geometrical meaning of our expansions in relation to quantum spectral curves
and $\beta$-ensembles of matrix models arising in refined topological string
theory.
