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

TL;DR

This paper introduces nonsymmetric interpolation Laurent polynomials in multiple variables, generalizing Okounkov's BC-type interpolation Macdonald polynomials, and explores their properties and conjectures on vanishing behavior.

Contribution

It presents a new class of nonsymmetric interpolation Laurent polynomials depending on parameters, connecting them to Macdonald polynomials via Hecke algebra symmetrization.

Findings

Recovery of Okounkov's BC-type polynomials through symmetrization

Introduction of parameter-dependent nonsymmetric Laurent polynomials

Conjectures on extra vanishing based on computational evidence

Abstract

Nonsymmetric interpolation Laurent polynomials in $n$ variables are introduced, with the interpolation points depending on $q$ and on a $n$-tuple of parameters $\tau=(\tau_1,\ldots,\tau_n)$. When $\tau_i=st^{n-i}$ Okounkov's $3$-parameter $BC_n$-type interpolation Macdonald polynomials are recovered from the nonsymmetric interpolation Laurent polynomials through Hecke algebra symmetrisation with respect to a type $C_n$ Hecke algebra action. In the appendix we give some conjectures about extra vanishing, based on Mathematica computations in rank two.