Nonsymmetric interpolation macdonald polynomials and $ \mathfrak{g}\mathfrak{l}_n $ basic hypergeometric series

Abstract

The Knop-Sahi interpolation Macdonald polynomials are inhomogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation Macdonald polynomials to study a new type of basic hypergeometric series of type gl n . Our main results include a new q-binomial theorem, a new q-Gauss sum, and several transformation formulae for gl n series.

Notes on Schubert, Grothendieck and Key Polynomials

“…. , x n−1 ), can be used to generate the interpolation Macdonald polynomials as well as the nonsymmetric Macdonald polynomials, see [45] for details.…”

“…The operators T i := T t,−t,1,0,0 i , 0 ≤ i ≤ n − 1 have been used in [45] to give an "elementary" construction of nonsymmetric Macdonald polynomials. Indeed, one can realize the operator π as follows:…”

“…, n − 2, and φ 2 T n−1 = T 1 φ 2 . It has been established in [45] how to use the operators φ, T 1 , . .…”

Multiple bracket function, Stirling number, and Lah number identities

“…This special case and its nonsymmetric analogues are also proved independently in [24,26], and studied in more recent works [21].…”

Clustering properties of rectangular Macdonald polynomials

“…Affine Hecke Algebra. We use the notation of Lascoux, Rains and Warnaar [22]. Let us recall it here.…”