By the fundamental theorem of symmetric polynomials, if P ∈ Q[X1, . . . , Xn] is symmetric, then it can be written P = Q(σ1, . . . , σn), where σ1, . . . , σn are the elementary symmetric polynomials in n variables, and Q is in Q[S1, . . . , Sn]. We investigate the complexity properties of this construction in the straight-line program model, showing that the complexity of evaluation of Q depen...

In this paper we introduce a family of polynomials indexed by pairs of partitions and show that if these polynomials are self–orthogonal then the centre of the Iwahori–Hecke algebra of the symmetric group is precisely the set of symmetric polynomials in the Murphy operators.

We prove a nonsymmetric analogue of a formula of Kato and Lusztig which describes the coefficients of the expansion of irreducible Weyl characters in terms of (degenerate) symmetric Macdonald polynomials as certain Kazhdan–Lusztig polynomials. We also establish precise polynomiality results for coefficients of symmetric and nonsymmetric Macdonald polynomials and a version of Demazure’s characte...

In this paper we introduce a family of polynomials indexed by pairs of partitions and show that if these polynomials are self-orthogonal then the centre of the Iwahori-Hecke algebra of the symmetric group is precisely the set of symmetric polynomials in the Murphy operators.

Background material on α-shift radix systems and α-CNS polynomials is collected. Symmetric CNS trinomials of the shape Xd + bX + c (d > 2) are characterized, thereby extending known results on quadratic symmetric CNS polynomials.

In this paper, we introduce the modified q-Euler polynomials. The main objective of this paper is to consider symmetric identities of the modified q-Euler polynomials under the symmetric group of degree n. AMS subject classification: 11B68, 11S40, 11S80.

A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of r...

Knop and Sahi simultaneously introduced a family of non-homogeneous, non-symmetric polynomials, Gα(x; q, t). The top homogeneous components of these polynomials are the non-symmetric Macdonald polynomials, Eα(x; q, t). An appropriate Hecke algebra symmetrization of Eα yields the Macdonald polynomials, Pλ(x; q, t). A search for explicit formulas for the polynomials Gα(x; q, t) led to the main re...

Jack polynomials are a remarkable family of polynomials in n variables x = (x1, · · · , xn) with coefficients in the field F := Q(α) where α is an indeterminate. They arise naturally in several statistical, physical, combinatorial, and representation theoretic considerations. The symmetric polynomials ([M1], [St], [LV], [KS]) Jλ = J (α) λ are indexed by partitions λ = (λ1, · · · , λn) where λ1 ...

In this paper we derive the bi-orthogonality relations, diagonal term evaluations and evaluation formulas for the non-symmetric Koornwinder polynomials. For the derivation we use certain representations of the (double) affine Hecke algebra which were originally defined by Noumi and Sahi. We furthermore give the explicit connection between the non-symmetric and the symmetric theory. This leads i...