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Regular version of the site

Article

Interpolation Macdonald polynomials and Cauchy-type identities

Journal of Combinatorial Theory, Series A. 2019. Vol. 162. P. 65-117.

Let Sym denote the algebra of symmetric functions and P_μ( · ; q, t) and Q_μ( · ; q, t) be the Macdonald symmetric functions (recall that they differ by scalar factors only). The (q, t)-Cauchy identity
expresses the fact that the P_μ( · ; q, t)’s form an orthogonal basis in Sym with respect to a special scalar product  (· , ·)depending on q and t. The present paper deals with the inhomogeneous interpolationMacdonald symmetric functions
I_μ(x_1, x_2, . . . ; q, t) = P_μ(x_1, x_2, . . . ; q, t)+lower degree terms.
These functions come from the N-variate interpolation Macdonald polynomials, extensively studied in the 90s by Knop, Okounkov, and Sahi. The goal of the paper is to construct symmetric functions H_μ( · ; q, t) with the biorthogonality property
(I_μ( · ; q, t),H_ν ( · ; q, t)) = δμν.