We define a set of orthogonal functions on the complex projective space CPN−1, and compute their Clebsch-Gordan coefficients as well as a large class of 6–j symbols. We also provide all the needed formulae for the generation of high-temperature expansions for U(N)-invariant spin models defined on CPN−1.

In this paper we use tournament matrices to give a combinatorial interpretation for the entries of the inverse t-Kostka matrix, which is the transition matrix between the Hall-Littlewood polyno-mials and the Schur functions. 0.1 Introduction In the rst section of this paper we introduce some basic notation about tournament matrices, and prove a theorem that is crucial in the second section. In ...

We report about some results, interesting examples, problems and conjectures revolving around the parabolic Kostant partition functions, the parabolic Kostka polynomials and “saturation” properties of several generalizations of the Littlewood–Richardson numbers.

If [λ(j)] is a multipartition of the positive integer n (a sequence of partitions with total size n), and μ is a partition of n, we study the number K[λ(j)]μ of sequences of semistandard Young tableaux of shape [λ(j)] and total weight μ. We show that the numbers K[λ(j)]μ occur naturally as the multiplicities in certain permutation representations of wreath products. The main result is a set of ...

We give several equivalent combinatorial descriptions of the space of states for the box-ball systems, and connect certain partition functions for these models with the q-weight multiplicities of the tensor product of the fundamental representations of the Lie algebra gl(n). As an application, we give an elementary proof of the special case t = 1 of the Haglund–Haiman–Loehr formula. Also, we pr...

This work studies the remarkable relationships that hold among certain m-tuples of the Garsia-Haiman modules M µ and corresponding elements of the Macdonald basis. We recall that in [10], M µ is defined for a partition µ ⊢ n, as the linear span of derivatives of a certain bihomogeneous polynomial ∆ µ y n. It has been conjectured in [6], [10] that M µ has n! dimensions and that its bigraded Frob...

Let Mw = (P)//SL2 denote the geometric invariant theory quotient of (P) by the diagonal action of SL2 using the line bundle O(w1, w2, . . . , wn) on (P). Let Rw be the coordinate ring of Mw. We give a closed formula for the Hilbert function of Rw, which allows us to compute the degree of Mw. The graded parts of Rw are certain Kostka numbers, so this Hilbert function computes stretched Kostka nu...