## Abstract

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 ∑μP_{μ}(x_{1},x_{2},…;q,t)Q_{μ}(y_{1},y_{2},…;q,t)=∏i,j=1∞[Formula presented] expresses the fact that the P_{μ}(⋅;q,t)'s form an orthogonal basis in Sym with respect to a special scalar product 〈⋅,⋅〉_{q,t}. The present paper deals with the inhomogeneous interpolation Macdonald 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)〉_{q,t}=δ_{μν}. These new functions live in a natural completion Symˆ⊃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 (q,t)=(q,q^{k})→(1,1) is also described.

Original language | English |
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Pages (from-to) | 65-117 |

Number of pages | 53 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 162 |

DOIs | |

Publication status | Published - Feb 2019 |

## Keywords

- Biorthogonal systems
- Cauchy identity
- Interpolation polynomials
- Jack polynomials
- Macdonald polynomials
- Symmetric functions