Yang-Baxter Graphs, Jack and Macdonald Polynomials

Abstract

We describe properties of the affine graph underlying the recursions between the different varieties of nonsymmetric Macdonald and Jack polynomials. We use an arbitrary function of one variable in the definition of affine edges, and of Cherednik's elements, to unify the different theories. We describe the symmetrizing operators furnishing the symmetric polynomials from the nonsymmetric ones.

Vector-Valued Jack Polynomials from Scratch

“…Following Lascoux [14] we define the monoid S N , a subsemigroup of the affine symmetric group, with generators {s 1 , s 2 , . .…”

Polynomial Representations of the Hecke Algebra of the Symmetric Group

“…We shall see in the next section that the graph Γ λ provides polynomials having the required vanishing conditions. In fact, Sahi [26] and Knop [14] use a similar construction to define non-symmetric non-homogeneous Macdonald polynomials, we have rephrased it in terms of Yang-Baxter graphs in [18]. [r]…”

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

“…, x n ). The definition of the interpolation Macdonald polynomial M u (x) = M u (x; q, t) is deceptively simple [10], [11], [13], [29], [31]. It is the unique polynomial of degree |u| such that…”

“…For a more general view of this recursive construction in terms of Yang-Baxter graphs we refer the reader to [11].…”

“…Various multivariable generalisations of the Newton interpolation polynomials exist in the literature, such as the Schubert polynomials [12] and several types of Macdonald interpolation polynomials [10], [11], [23], [24], [25], [29], [31]. In this paper we are interested in the latter, providing generalisations of (1.1) when the interpolation points x 1 , .…”