On factorization of generalized Macdonald polynomials

Ya Kononov,A Morozov

doi:10.1140/epjc/s10052-016-4276-5

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.

Highlights

In this letter we address one of them—what happens to the hook formulas for classical, quantum, and Macdonald dimensions at the level of generalized Macdonald polynomials (GMPs)? We find that they survive, but only partly—on a onedimensional line in the space of time variables
Coarm a and coleg l are the ordinary coordinates of the box in the diagram
The GMP depends on a set of Young diagrams and a set of time variables—in the following we consider the simplest non-trivial case, when there are two: two Young diagrams and two sets of times

Summary

Introduction

Coarm a and coleg l are the ordinary coordinates of the box in the diagram. To keep the notation consistent throughout the text, in (1) we call the relevant parameter t, not q, from the very beginning. By a factorization of S( f ) we mean that its plethystic logarithm f is a polynomial or, more generally, a rational function, with integer coefficients. The quantities χ are deformations of the dimensions of representation R of S L N -algebras, which factorize due to Weyl formulas, and are called quantum and Macdonald dimensions.

Results

Conclusion