We establish a connection between a specialization of the nonsymmetric Macdonald polynomials and the Demazure characters of the corresponding affine Kac-Moody algebra. This allows us to obtain a representation-theoretical interpretation of the coefficients of the expansion of the specialized symmetric Macdonald polynomials in the basis formed by the irreducible characters of the associated fini...

The Askey-Wilson polynomials are orthogonal polynomials in x = cos θ, which are given as a terminating 4φ3 basic hypergeometric series. The non-symmetric AskeyWilson polynomials are Laurent polynomials in z = eiθ, which are given as a sum of two terminating 4φ3’s. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single 4φ3’s which are Laurent polynomials in...

We give two examples where symmetric polynomials play an important rôle in physics: First, the partition functions of ideal quantum gases are closely related to certain symmetric polynomials, and a part of the corresponding theory has a thermodynamical interpretation. Further, the same symmetric polynomials also occur in Berezin’s theory of quantization of phase spaces with constant curvature.

We give a combinatorial formula for the non-symmetric Macdonald polynomials E µ (x; q, t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J µ (x; q, t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop, that characterizes the non-symmetric Macdonald polynomials.

We give a combinatorial formula for the non-symmetric Macdonald polynomials E µ (x; q, t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J µ (x; q, t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop and Sahi, that characterizes the non-symmetric Macdonald polynomials.

We give a combinatorial formula for the non-symmetric Macdonald polynomials Eμ(x; q, t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials Jμ(x; q, t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop and Sahi, that characterizes the non-symmetric Macdonald polynomials.

We give a combinatorial formula for the non-symmetric Macdonald polynomials E µ (x; q, t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J µ (x; q, t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop and Sahi, that characterizes the non-symmetric Macdonald polynomials.

We find a biorthogonal expansion of the Cayley transform of the non-symmetric Jack functions in terms of the non-symmetric Jack polynomials, the coefficients being Meixner–Pollaczek type polynomials. This is done by computing the Cherednik–Opdam transform of the non-symmetric Jack polynomials multiplied by the exponential function.

We revisit a geometric lower bound for Waring rank of polynomials (symmetric rank of symmetric tensors) of [LT10] and generalize it to a lower bound for rank with respect to arbitrary varieties, improving the bound given by the “non-Abelian” catalecticants recently introduced by Landsberg and Ottaviani. This is applied to give lower bounds for ranks of multihomogeneous polynomials (partially sy...