We give half a dozen bases of the Hecke algebra of the symmetric group, and relate them to the basis of Geck-Rouquier, and to the basis of Jones, using matrices of change of bases of the ring of symmetric polynomials.

The symmetric algebra of a (finite-dimensional) g-module V is the algebra of polynomial functions on the dual space V . Therefore one can study the algebra of symmetric invariants using geometry of G-orbits in V ∗ . In case of the exterior algebra, ∧•V, lack of such geometric picture results by now in absence of general structure theorems for the algebra of skew-invariants (∧•V)g . One may find...

In this article, we carry out the investigation for regular sequences of symmetric polynomials in the polynomial ring in three and four variable. Any two power sum element in C[x1,x2, . . . ,xn] for n ≥ 3 always form a regular sequence and we state the conjecture when pa, pb, pc for given positive integers a < b < c forms a regular sequence in C[x1,x2,x3,x4]. We also provide evidence for this c...

Structure constants for the multiplication of Schubert polynomials by Schur symmetric polynomials are related to the enumeration of chains in a new partial order on S 1 , the Grassmaniann Bruhat order. Here we present a monoid M related to this order. We develop a notion of reduced sequences for M and show that M is analogous to the nil-Coxeter monoid for the weak order on S 1 .

In this paper, we consider a family of symmetric polynomials of the eigenvalues of a complex matrix A and find an explicit expression of each member of the family as a polynomial of the entries of A with positive coefficients. In the case of a nonnegative matrix, one immediately obtains a family of inequalities involving matrix eigenvalues and diagonal entries. Equivalent forms of some of the o...

Let wλ x : 1 − x2 λ−1/2 and Pλ,n x be the ultraspherical polynomials with respect to wλ x . Then, we denote the Stieltjes polynomials with respect to wλ x by Eλ,n 1 x satisfying ∫1 −1 wλ x Pλ,n x Eλ,n 1 x x dx 0, 0 ≤ m < n 1, ∫1 −1 wλ x Pλ,n x Eλ,n 1 x xdx / 0, m n 1. In this paper, we investigate asymptotic properties of derivatives of the Stieltjes polynomials Eλ,n 1 x and the product Eλ,n 1 ...

By means of comp lete symmetric polynomials this paper gives a new proof for the Vander-monde determinant formula. Another alternative proof for this formula is obtained via the collocatio n matrices. It also gives a generalized relationship between the Vandermo nde, the Pascal and the Stirling matrices. A new app roach to obtain the explicit inverse of the Vandermo nde matrix is investigated. ...

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.

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...

Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials. We also compute the canonical invariants for all dihedral groups as certain hypergeometric functions.