We present two symmetric function operators H 3 and H qt 4 that have the property H mH(2a1b)(X; q, t) = H(m2a1b)(X; q, t). These operators are generalizations of the analogous operator H 2 and also have expressions in terms of Hall-Littlewood vertex operators. We also discuss statistics, aμ(T ) and bμ(T ), on standard tableaux such that the q, t Kostka polynomials are given by the sum over stan...

Kostka-Folkes polynomials may be considered as coefficients of the formal power series representing the character of certain graded GL(n)-modules. These GL(n)-modules are defined by twisting the coordinate ring of the nullcone by a suitable line bundle [1] and the definition may be generalized by twisting the coordinate ring of any nilpotent conjugacy closure in gl(n) by a suitable vector bundl...

i=1 ti−1 (1+· · ·+qμi−1). In [8] Garsia-Tesler proved that if γ is a partition of k and λ = (n−k, γ) is a partition of n, then there is a unique symmetric polynomial kγ(x; q, t) of degree ≤ k with the property that K̃λμ(q, t) = kγ [Bμ(q, t); q, t] holds true for all partitions μ. It was shown there that these polynomials have Schur function expansions of the form kγ(x; q, t) = ∑ |ρ|≤|γ| Sλ(x) kρ...

Stanley has studied a symmetric function generalization XG of the chromatic polynomial of a graph G. The innocent-looking Stanley-Stembridge Poset Chain Conjecture states that the expansion of XG in terms of elementary symmetric functions has nonnegative coefficients if G is a clawfree incomparability graph. Here we give a combinatorial interpretation of these coefficients by combining Gasharov...

We construct for each μ ` n a bigraded Sn-module Hμ and conjecture that its Frobenius characteristic Cμ(x; q, t) yields the Macdonald coefficients Kλμ(q, t). To be precise, we conjecture that the expansion of Cμ(x; q, t) in terms of the Schur basis yields coefficients Cλμ(q, t) which are related to the Kλμ(q, t) by the identity Cλμ(q, t) = Kλμ(q, 1/t)t. The validity of this would give a represe...

Every symmetric function f can be written uniquely as a linear combination of Schur functions, say f = ∑ λ xλsλ, and also as a linear combination of fundamental quasisymmetric functions, say f = ∑ α yαQα. For many choices of f arising in the theory of Macdonald polynomials and related areas, one knows the quasisymmetric coefficients yα and wishes to compute the Schur coefficients xλ. This paper...

The Kostka numbers Kλμ are important in several areas of mathematics, including symmetric function theory, representation theory, combinatorics and invariant theory. The q-Kostka polynomials Kλμ(q) are the q-analogues of the Kostka numbers. They generalize and extend the mathematical meaning of the Kostka numbers. Lascoux and Schützenberger proved one can attach a non-negative integer statistic...