On Factorization of Generalized Macdonald Polynomials

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 $W_\infty$ -- 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 $U_q(SL_N)$ 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.