A nonsymmetric version of Okounkov's BC-type interpolation Macdonald polynomials

Symmetric and nonsymmetric interpolation Laurent polynomials are introduced with the interpolation points depending on $q$ and a $n$-tuple of parameters $\tau=(\tau_1,\ldots,\tau_n)$. For the principal specialization $\tau_i=st^{n-i}$ the symmetric interpolation Laurent polynomials reduce to Okounkov's $BC$-type interpolation Macdonald polynomials and the nonsymmetric interpolation Laurent polynomials become their nonsymmetric variants. We expand the symmetric interpolation Laurent polynomials in the nonsymmetric ones. We show that Okounkov's $BC$-type interpolation Macdonald polynomials can also be obtained from their nonsymmetric versions using a one-parameter family of actions of the finite Hecke algebra of type $B_n$ in terms of Demazure-Lusztig operators. In the Appendix we give some experimental results and conjectures about extra vanishing.Comment: 30 pages, 9 figures; v4: experimental results and conjectures added about extra vanishin

Some remarks on non-symmetric interpolation Macdonald polynomials

We provide elementary identities relating the three known types of non-symmetric interpolation Macdonald polynomials. In addition we derive a duality for non-symmetric interpolation Macdonald polynomials. We consider some applications of these results, in particular for binomial formulas involving non-symmetric interpolation Macdonald polynomials.Comment: 20 pages; v2: typos corrected; v3: final version with minor corrections. To appear in IMR

BC_n-symmetric polynomials

We consider two important families of BC_n-symmetric polynomials, namely Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials. We give a family of difference equations satisfied by the former, as well as generalizations of the branching rule and Pieri identity, leading to a number of multivariate q-analogues of classical hypergeometric transformations. For the latter, we give new proofs of Macdonald's conjectures, as well as new identities, including an inverse binomial formula and several branching rule and connection coefficient identities. We also derive families of ordinary symmetric functions that reduce to the interpolation and Koornwinder polynomials upon appropriate specialization. As an application, we consider a number of new integral conjectures associated to classical symmetric spaces.Comment: 65 pages, LaTeX. v2-3: Minor corrections and additions (including teasers for the sequel). v4: C^+ notation changed to harmonize with the sequels (and more teasers added

Interpolation with bilinear differential forms

We present a recursive algorithm for modeling with bilinear differential forms. We discuss applications of this algorithm for interpolation with symmetric bivariate polynomials, and for computing storage functions for autonomous systems

Harmonic functions on multiplicative graphs and interpolation polynomials

We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur's S and P functions and with Jack symmetric functions. As a by-product, we compute certain Selberg-type integrals.Comment: AMSTeX, 35 page

Sparse multivariate polynomial interpolation in the basis of Schubert polynomials

Schubert polynomials were discovered by A. Lascoux and M. Sch\"utzenberger in the study of cohomology rings of flag manifolds in 1980's. These polynomials generalize Schur polynomials, and form a linear basis of multivariate polynomials. In 2003, Lenart and Sottile introduced skew Schubert polynomials, which generalize skew Schur polynomials, and expand in the Schubert basis with the generalized Littlewood-Richardson coefficients. In this paper we initiate the study of these two families of polynomials from the perspective of computational complexity theory. We first observe that skew Schubert polynomials, and therefore Schubert polynomials, are in \CountP (when evaluating on non-negative integral inputs) and \VNP. Our main result is a deterministic algorithm that computes the expansion of a polynomial $f$ of degree $d$ in $\Z[x_1, \dots, x_n]$ in the basis of Schubert polynomials, assuming an oracle computing Schubert polynomials. This algorithm runs in time polynomial in $n$, $d$, and the bit size of the expansion. This generalizes, and derandomizes, the sparse interpolation algorithm of symmetric polynomials in the Schur basis by Barvinok and Fomin (Advances in Applied Mathematics, 18(3):271--285). In fact, our interpolation algorithm is general enough to accommodate any linear basis satisfying certain natural properties. Applications of the above results include a new algorithm that computes the generalized Littlewood-Richardson coefficients.Comment: 20 pages; some typos correcte