On Factorization of Generalized Macdonald Polynomials

Abstract

A remarkable feature of Schur functions -- the common eigenfunctions of cut-and-join operators from $W_{\infty }$ -- is that they factorize at the peculiar two-parametric topological locus in the space of time-variables, what is known as the hook formula for quantum dimensions of representations of $U_q(S{L}_N)$ and plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to {\it generalized} Macdonald polynomials (GMP), 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 codimension-one slice of the topological locus, what is already a very non-trivial property, calling for proof and better understanding.

8 pages

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• Cut-and-join operators and Macdonald polynomials from the 3-Schur functions
• Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed