Interpolation Macdonald polynomials and Cauchy-type identities

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 $$\sum _{\mu }{P}_{\mu }(x_1,x_2,\dots ;q,t)Q_{\mu }(y_1,y_2,\dots ;q,t)=\prod _{i,j=1}^{\infty }\frac{(x_i{y}_{j}t;q{)}_{\infty }}{(x_i{y}_j;q{)}_{\infty }}$$ expresses the fact that the $P_{\mu }(\cdot ;q,t)$'s form an orthogonal basis in Sym with respect to a special scalar product $⟨\cdot ,\cdot {⟩}_{q,t}$. 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 $$⟨I_{\mu }(\cdot ;q,t),J_{\nu }(\cdot ;q,t){⟩}_{q,t}={\delta }_{\mu \nu }.$$ 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.

v2: the notation of dual functions changed in accordance with reviewer's suggestion; a few typos fixed; three unnecessary references deleted from the list; to appear in J. Combin. Theory Ser. A

References in corpus (1)

• Lectures on Integrable probability: Stochastic vertex models and symmetric functions

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