A short proof of generalized Jacobi-Trudi expansions for Macdonald polynomials

Based on a generalized Newton's identity, we construct a family of symmetric functions which deform the modular Hall-Littlewood functions. We also give a determinant formula for the Macdonald functions.

In symmetric Macdonald polynomial theory the Pieri formula gives the branching coefficients for the product of the rth elementary symmetric function and the Macdonald polynomial. In this paper we give the nonsymmetric analogues for the cases r=1 and r=n-1. We do this by first deducing the the decomposition for the product of any nonsymmetric Macdonald polynomial with a linear function in terms of nonsymmetric Macdonald polynomials. As a corollary of finding the branching coef...

In the 90's a collection of Plethystic operators were introduced in [3], [7] and [8] to solve some Representation Theoretical problems arising from the Theory of Macdonald polynomials. This collection was enriched in the research that led to the results which appeared in [5], [6] and [9]. However since some of the identities resulting from these efforts were eventually not needed, this additional work remained unpublished. As a consequence of very recent publications [4], [11...

In this paper, we introduce higher rank generalizations of Macdonald polynomials. The higher rank non-symmetric Macdonald polynomials are Laurent polynomials in several sets of variables which form weight bases for higher rank polynomial representations of double affine Hecke algebras with respect to higher rank Cherednik operators. We prove that these polynomials satisfy generalized versions of the classical Knop--Sahi relations and we give combinatorial descriptions of thei...

We present a new formula of Cauchy type for the nonsymmetric Macdonald polynomials which are joint eigenfunctions of q-Dunkl operators. This gives the explicit formula for a reproducing kernel on the polynomial ring of n variables.

Generalized Macdonald polynomials (GMP) are eigenfunctions of specifically-deformed Ruijsenaars Hamiltonians and are built as triangular polylinear combinations of Macdonald polynomials. They are orthogonal with respect to a modified scalar product, which could be constructed with the help of an increasingly important triangular perturbation theory, showing up in a variety of applications. A peculiar feature of GMP is that denominators in this expansion are fully factorized, ...

Multiple-integral representations of the (skew-)Macdonald symmetric functions are obtained. Some bosonization schemes for the integral representations are also constructed.

This paper is a continuation of our papers [EK1, EK2]. In [EK2] we showed that for the root system A_n-1 one can obtain Macdonald's polynomials - a new interesting class of symmetric functions recently defined by I. Macdonald {M1] - as weighted traces of intertwining operators between certain finite-dimensional representations of U_q sl_n. The main goal of the present paper is to use this construction to give a representation-theoretic proof of Macdonald's inner product and s...

We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases $t=1$ and $q=0$, we recover known expressions for the monomial symmetric and Hall-Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and $q$-Whittaker polynomials.

We study Macdonald polynomials from a basic hypergeometric series point of view. In particular, we show that the Pieri formula for Macdonald polynomials and its recently discovered inverse, a recursion formula for Macdonald polynomials, both represent multivariable extensions of the terminating very-well-poised 6-phi-5 summation formula. We derive several new related identities including multivariate extensions of Jackson's very-well-poised 8-phi-7 summation. Motivated by our...