Kostka numbers and Littlewood-Richardson coefficients appear in combinatorics and representation theory. Interest in their computation stems from the fact that they are present in quantum mechanical computations since Wigner [15]. In recent times, there have been a number of algorithms proposed to perform this task [1–3, 11, 12]. The issue of their computational complexity has received attentio...

In [11], Haiman proved the remarkable result that the isospectral Hilbert scheme of points in the plane is normal, Cohen-Macaulay and Gorenstein. He also showed that this implies the n! conjecture and the positivity conjecture for the Kostka-Macdonald coefficients. In addition, he conjectured that the isospectral Hilbert scheme over the principal component of Hilb(Cd) is Cohen-Macaulay for any ...

We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Litt...

We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion of Vakil and a special position argument due to Schubert, our result follows from a particular inequality among Kostka numbers of two-rowed tableau...

Sylwia Pośpiech-Kurkowska, Paweł Kostka, Arkadiusz Gertych, Ewa Straszecka, Joanna Straszecka* Div. of Biomedical Engineering, Institute of Electronics, Silesian Technical University 16 Akademicka St. 44-100 Gliwice, Poland phone +48 (32) 2371725, fax +48 (32) 2372235, e-mail: {sylwia,kostka,gertych,ewa} @biomed.iele.polsl.gliwice.pl *Department of Internal Diseases and Rheumatology, Medical Un...

A graded poset structure is defined for the sets of LittlewoodRichardson (LR) tableaux that count the multiplicity of an irreducible gl(n)module in the tensor product of irreducible gl(n)-modules corresponding to rectangular partitions. This poset generalizes the cyclage poset on columnstrict tableaux defined by Lascoux and Schützenberger, and its grading function generalizes the charge statist...

The k-Young lattice Y k is a partial order on partitions with no part larger than k. This weak subposet of the Young lattice originated [9] from the study of the k-Schur functions s (k) λ , symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed by k-bounded partitions. The chains in the k-Young lattice are induced by a Pieri-type rule experimentally ...

We explain how beautiful combinatorial constructions involving the Robinson-Schensted-Knuth correspondence, evacuation of tableaux, and the Kostka-Foulkes polynomials, arise naturally from the structure of (affine) crystal graphs. The appearance of Kostka-Foulkes polynomials was observed by Nakayashiki and Yamada. Almost all of the constructions presented herein, have analogues for every simple...

We present two symmetric function operators H qt 3 and H qt 4 that have the property H qt 3 H (2 a 1 b) [X; q, t] = H (32 a 1 b) [X; q, t] and H qt 4 H (2 a 1 b) [X; q, t] = H (42 a 1 b) [X; q, t]. These operators are generalizations of the analogous operator H qt 2 and also have expressions in terms of Hall-Littlewood vertex operators. We also discuss statistics, a µ (T) and b µ (T), on standa...