Kernel function and quantum algebras

TL;DR

This paper introduces a new kernel function related to Macdonald polynomials, explores its algebraic properties, and connects it to quantum algebra representations and deformed $ ext{W}$ algebra correlations.

Contribution

It constructs an analogue of the Cauchy kernel for Macdonald polynomials within a tensor product framework and links it to quantum algebra representations and correlation functions.

Findings

The kernel $K_n(x,z;q,t)$ is described by the tableau sum formula.

The integer level representation of Ding-Iohara algebra produces deformed $ ext{W}$ algebra currents.

The kernel appears in the highest-to-highest correlation functions of the deformed $ ext{W}$ algebra.

Abstract

We introduce an analogue $K_n(x,z;q,t)$ of the Cauchy-type kernel function for the Macdonald polynomials, being constructed in the tensor product of the ring of symmetric functions and the commutative algebra $\mathcal{A}$ over the degenerate $\mathbb{C} \mathbb{P}^1$. We show that a certain restriction of $K_n(x,z;q,t)$ with respect to the variable $z$ is neatly described by the tableau sum formula of Macdonald polynomials. Next, we demonstrate that the integer level representation of the Ding-Iohara quantum algebra naturally produces the currents of the deformed $\mathcal{W}$ algebra. Then we remark that the $K_n(x,z;q,t)$ emerges in the highest-to-highest correlation function of the deformed $\mathcal{W}$ algebra.