On factorization of generalized Macdonald polynomials
A remarkable feature of Schur functions—the common eigenfunctions of cut-and-join operators from W∞W∞—is that they factorize at the peculiar two-parametric topological locus in the space of time variables, which is known as the hook formula for quantum dimensions of representations of Uq(SLN)Uq(SLN) and which plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to generalized Macdonald polynomials (GMPs), associated in the same way with the toroidal Ding–Iohara–Miki algebras, which play the central role in modern studies in Seiberg-Witten–Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time variables, we discover a weak factorization—on a one- (rather than four-) parametric slice of the topological locus, which is already a very non-trivial property, calling for proof and better understanding.