On Factorization of Generalized Macdonald Polynomials
Ya. Kononov, A. Morozov

TL;DR
This paper explores the factorization properties of generalized Macdonald polynomials related to advanced algebraic structures, revealing a weak factorization on a specific locus, which extends known properties of classical symmetric functions.
Contribution
It introduces a new weak factorization property of generalized Macdonald polynomials on a special locus, extending classical factorization results to a broader algebraic context.
Findings
Discovered weak factorization of GMP on a codimension-one slice
Extended classical hook formula to generalized Macdonald polynomials
Identified a non-trivial property requiring further proof and understanding
Abstract
A remarkable feature of Schur functions -- the common eigenfunctions of cut-and-join operators from -- 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 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…
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