For certain negative rational numbers κ0, called singular values, and associated with the symmetric group SN on N objects, there exist homogeneous polynomials annihilated by each Dunkl operator when the parameter κ = κ0. It was shown by de Jeu, Opdam and the author (Trans. Amer. Math. Soc. 346 (1994), 237-256) that the singular values are exactly the values − n with 2 ≤ n ≤ N , m = 1, 2, . . . ...

We give an algebro-combinatorial proof of a general version of Pieri’s formula following the approach developed by Fomin and Kirillov in the paper “Quadratic algebras, Dunkl elements, and Schubert calculus.” We prove several conjectures posed in their paper. As a consequence, a new proof of classical Pieri’s formula for cohomology of complex flag manifolds, and that of its analogue for quantum ...

1. Signature Characters and the Jantzen Filtration 1 1.1. Jantzen Filtrations on Standard Modules 1 1.2. Hermitian Duals 3 1.3. Rationality of Signature Characters 5 1.4. Asymptotic Signatures 7 1.5. Isotypic Signature Characters 8 2. The Dunkl Weight Function 10 2.1. The Gaussian Inner Product 10 2.2. Weight Function 10 2.3. Analytic Continuation on the Regular Locus 20 2.4. Extension to Distr...

The purpose of this paper is to introduce a modification of q-Dunkl generalization of exponential functions. These types of operators enable better error estimation on the interval [Formula: see text] than the classical ones. We obtain some approximation results via a well-known Korovkin-type theorem and a weighted Korovkin-type theorem. Further, we obtain the rate of convergence of the operato...

We suggest a new combinatorial construction for the cohomology ring of the ag manifold. The degree 2 commutation relations satissed by the divided diierence operators corresponding to positive roots deene a quadratic associative algebra. In this algebra, the formal analogues of Dunkl operators generate a commuta-tive subring, which is shown to be canonically isomorphic to the cohomology of the ...