this paper is to add to the existing characterizations of Jack polynomials two further ones: c) a recursion formula among the F together with two formulas to obtain J from them. d) combinatorial formulas of both J and F in terms of certain generalized tableaux

There is an algebra of commutative differential-difference operators which is very useful in studying analytic structures invariant under permutation of coordinates. This algebra is generated by the Dunkl operators T i := 1\Gamma(ij) x i \Gammax j , (i = 1; : : : ; N , where (ij) denotes the transposition of the variables x i ; x j and k is a fixed parameter). We introduce a family of functions fp ff g, indexed by m-tuples of non-negative integers ff = (ff 1 ; : : : ; am ) for m N , which allow a workable treatment of important constructions such as the intertwining operator V . This is a linear map on polynomials, preserving the degree of homogeneity, for which T i V = V ; i = 1; : : : ; N , normalized by V 1 = 1 (see Dunkl, Canadian J. Math. 43(1991), 1213--1227). We show that T i p ff = 0 for i ? m, and 1 ! 2 ! \Delta \Delta \Delta m ! A fiff p fi ; where ( 1 ; 2 ; : : : ; m ) is the partition whose parts are the entries of ff (That is, 1 2 m 0), fi = (fi 1 ; : : : ; fi m ); i=1 ff i and the sorting of fi is a partition strictly larger than in the dominance order. This triangular matrix representation of V allows a detailed study. There is an inner product structure on spanfp ff g and a convenient set of self-adjoint operators, namely T i ae i , where ae i p ff := p (ff 1 ;:::;ff i +1;:::;ff m) . This structure has a bi-orthogonal relationship with the Jack polynomials in m variables. Values of k for which V fails to exist are called singular values and were studied by de Jeu, Opdam, and the author in Trans. Amer. Math. Soc. 346(1994), 237--256. As a partial verification of a conjecture made in that paper, we construct, for any a = 1; 2; 3; : : : such that gcd(N \Gamma m + 1; a) ! (N \Gamma m + 1)=m and m N=2, a space of polynomials ...

this paper is study certain q-difference raising operators for Macdonald polynomials (of type A n\Gamma1 ) which are originated from the q-difference - reflection operators introduced in our previous paper [KN]. These operators can be regarded as a q-difference version of the raising operators for Jack polynomials introduced by L. Lapointe and L. Vinet [LV1, LV2]. As an application of our q-difference raising operators, we will give a proof of the integrality of the double Kostka coefficients which had been conjectured by I.G. Macdonald [Ma], Chapter VI. We will also determine their quasi-classical limits, which give rise to (differential) raising operators for Jack polynomials. (See also Notes at the end of

We present formulas of Rodrigues type giving the Macdonald polynomials for arbitrary partitions through the repeated application of creation operators B k , k = 1; : : : ; `() on the constant 1. Three expressions for the creation operators are derived one from the other. When the last of these expressions is used, the associated Rodrigues formula readily implies the integrality of the (q; t)-Kostka coefficients. The proofs given in this paper rely on the connection between affine Hecke algebras and Macdonald polynomials.

1.1. Partitions, generalized Vandermonde determinants and representation theory. 5 1.2. The Frobenius character formulae. 8

Formulas of Rodrigues-type for the Macdonald polynomials are presented. They involve creation operators, certain properties of which are proved and other conjectured.

There are examples of Calogero-Sutherland models associated to the Weyl groups of type A and B. When exchange terms are added to the Hamiltonians the systems have nonsymmetric eigenfunctions, which can be expressed as products of the ground state with members of a family of orthogonal polynomials. These polynomials can be defined and studied by using the differential-difference operators introduced by the author in Trans. Amer. Math. Soc. 1989 (311), 167-183. After a description of known results, particularly from the works of Baker and Forrester, and Sahi; there is a study of polynomials which are invariant or alternating for parabolic subgroups of the symmetric group. The detailed analysis depends on using two bases of polynomials, one of which transforms monomially under group actions and the other one is orthogonal. There are formulas for norms and point-evaluations which are simplifications of those of Sahi. For any parabolic subgroup of the symmetric group there is a skew operator on polynomials which leads to evaluation at (1,1,...,1) of the quotient of the unique skew polynomial in a given irreducible subspace by the minimum alternating polynomial, analogously to a Weyl character formula. The last section concerns orthogonal polynomials for the type B Weyl group with an emphasis on the Hermite-type polynomials. These can be expressed by using the generalized binomial coefficients. A complete basis of eigenfunctions of Yamamoto’s BN spin Calogero model is obtained by multiplying these polynomials by the ground state. 1

A new generalization of the Jack polynomials that incorporates fermionic variables is presented. These Jack superpolynomials are constructed as those eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland (CMS) model that decomposes triangularly in terms of the symmetric monomial superfunctions. Many explicit examples are displayed. Furthermore, various new results have been obtained for the supersymmetric version of the CMS models: the Lax formulation, the construction of the Dunkl operators and the explicit expressions for the conserved charges. The reformulation of the models in terms of the exchange-operator formalism is a crucial aspect of our analysis.

In 1977 G.P. Thomas has shown that the sequence of Schur polynomials associated to a partition can be comfortabely generated from the sequence of variables x = (x1 ; x2 ; x3 ; : : : ) by the application of a mixed shift/multiplication operator, which in turn can be easily computed from the set SY T () of standard Young tableaux of shape .

There is a commutative parametrized algebra of differential-difference operators associated to the Weyl group of type BN ; the group generated by sign-changes and permutations of the coordinates in R^N. The operators in the title intertwine the algebra generated by the partial derivatives with the aforementioned one. The main results of this paper are the construction of an ordered basis of polynomials on which the intertwining operator acts as a triangular matrix, the determination of the diagonal entries, and a closed form for the L²-norm of alternating polynomials for reflection groups with respect to the Macdonald-Selberg measure.