Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
11016749 | Journal of Combinatorial Theory, Series A | 2019 | 53 Pages |
Abstract
Let Sym denote the algebra of symmetric functions and Pμ(â
;q,t) and Qμ(â
;q,t) be the Macdonald symmetric functions (recall that they differ by scalar factors only). The (q,t)-Cauchy identityâμPμ(x1,x2,â¦;q,t)Qμ(y1,y2,â¦;q,t)=âi,j=1â(xiyjt;q)â(xiyj;q)â expresses the fact that the Pμ(â
;q,t)'s form an orthogonal basis in Sym with respect to a special scalar product ãâ
,â
ãq,t. The present paper deals with the inhomogeneous interpolation Macdonald symmetric functionsIμ(x1,x2,â¦;q,t)=Pμ(x1,x2,â¦;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(â
;q,t) with the biorthogonality propertyãIμ(â
;q,t),Hν(â
;q,t)ãq,t=δμν. These new functions live in a natural completion SymËâSym. 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)â(1,1) is also described.
Related Topics
Physical Sciences and Engineering
Mathematics
Discrete Mathematics and Combinatorics
Authors
Grigori Olshanski,