Interpolation Macdonald polynomials and Cauchy-type identities

TL;DR

This paper introduces new symmetric functions biorthogonal to interpolation Macdonald polynomials, leading to novel Cauchy-type identities and their Jack limit degeneration, enriching the theory of symmetric functions.

Contribution

It constructs biorthogonal symmetric functions to interpolation Macdonald polynomials and derives new Cauchy-type identities, extending the understanding of symmetric functions and their orthogonality properties.

Findings

Constructed biorthogonal symmetric functions $J_ u(\,ullet\,;q,t)$.

Derived new Cauchy-type identities involving interpolation Macdonald polynomials.

Described the degeneration of identities in the Jack limit.

Abstract

Let Sym denote the algebra of symmetric functions and $P_\mu(\,\cdot\,;q,t)$ and $Q_\mu(\,\cdot\,;q,t)$ be the Macdonald symmetric functions (recall that they differ by scalar factors only). The $(q,t)$-Cauchy identity $$ \sum_\mu P_\mu(x_1,x_2,\dots;q,t)Q_\mu(y_1,y_2,\dots;q,t)=\prod_{i,j=1}^\infty\frac{(x_iy_jt;q)_\infty}{(x_iy_j;q)_\infty} $$ expresses the fact that the $P_\mu(\,\cdot\,;q,t)$'s form an orthogonal basis in Sym with respect to a special scalar product $\langle\,\cdot\,,\,\cdot\,\rangle_{q,t}$. The present paper deals with the inhomogeneous \emph{interpolation} Macdonald symmetric functions $$ I_\mu(x_1,x_2,\dots;q,t)=P_\mu(x_1,x_2,\dots;q,t)+\text{lower degree terms}. $$ These functions come from the $N$-variate interpolation Macdonald polynomials, extensively studied in the 90's by Knop, Okounkov, and Sahi. The goal of the paper is to construct symmetric functions $J_\mu(\,\cdot\,;q,t)$ with the biorthogonality property $$ \langle I_\mu(\,\cdot\,;q,t), J_\nu(\,\cdot\,;q,t)\rangle_{q,t}=\delta_{\mu\nu}. $$ These new functions live in a natural completion of the algebra Sym. As a corollary one obtains a new Cauchy-type identity in which the interpolation Macdonald polynomials are paired with certain multivariate rational symmetric functions. The degeneration of this identity in the Jack limit is also described.