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$W$-representation realizes partition functions by an action of acut-and-join-like operator on the vacuum state with a zero-mode background. Weprovide explicit formulas of this kind for $\beta$- and $q,t$-deformations ofthe simplest rectangular complex matrix model. In the latter case, instead ofthe complicated definition in terms of multiple Jackson integrals, we definepartition functions as the weight-two series, made from Macdonald polynomials,which are evaluated at different loci in the space of time variables. Resultingexpression for the $\hat W$ operator appears related to the problem of simpleHurwitz numbers (contributing are also the Young diagrams with all but onelines of length two and one). This problem is known to exhibit niceintegrability properties. Still the answer for $\hat W$ can seem unexpectedlysophisticated and calls for improvements. Since matrix models lie at the verybasis of all gauge- and string-theory constructions, our exercise provides agood illustration of the jump in complexity between $\beta$- and$q,t$-deformations -- which is not always seen at the accidently simple levelof Calogero-Ruijsenaars Hamiltonians (where both deformations are equallystraightforward). This complexity is, however, quite familiar in the theoriesof network models, topological vertices and knots.

On $W$-representations of $\beta$- and $q,t$-deformed matrix models