Interpolation Macdonald polynomials and Cauchy-type identities

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3 Citations (Scopus)

Abstract

Let Sym denote the algebra of symmetric functions and Pμ(⋅;q,t) and Qμ(⋅;q,t) be the Macdonald symmetric functions (recall that they differ by scalar factors only). The (q,t)-Cauchy identity ∑μPμ(x1,x2,…;q,t)Qμ(y1,y2,…;q,t)=∏i,j=1∞[Formula presented] expresses the fact that the Pμ(⋅;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 Iμ(x1,x2,…;q,t)=Pμ(x1,x2,…;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 Hμ(⋅;q,t) with the biorthogonality property 〈Iμ(⋅;q,t),Hν(⋅;q,t)〉q,tμν. 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,qk)→(1,1) is also described.

Original languageEnglish
Pages (from-to)65-117
Number of pages53
JournalJournal of Combinatorial Theory. Series A
Volume162
DOIs
Publication statusPublished - Feb 2019

Keywords

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

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