This article uses combinatorial objects called labeled abaci to give direct combinatorial proofs of many familiar facts about Schur polynomials. We use abaci to prove the Pieri rules, the Littlewood–Richardson rule, the equivalence of the tableau definition and the determinant definition of Schur polynomials, and the combinatorial interpretation of the inverse Kostka matrix (first given by Eğec...

Given a positive integer n, and partitions λ and μ of n, let Kλμ denote the Kostka number, which is the number of semistandard Young tableaux of shape λ and weight μ. Let J(λ) denote the number of μ such that Kλμ = 1. By applying a result of Berenshtein and Zelevinskii, we obtain a formula for J(λ) in terms of restricted partition functions, which is recursive in the number of distinct part siz...

We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron-Garsia nabla operator. The formula involves nested labelled Dyck paths weighted by area and a suitable “diagonal inversion” statistic. Our model includes as special cases many previous conjectures connecting the nabla operator to quantum lattice paths. The combinatorics of the ...

We study the orbits of G = GL(V ) in the enhanced nilpotent cone V ×N , where N is the variety of nilpotent endomorphisms of V . These orbits are parametrized by bipartitions of n = dimV , and we prove that the closure ordering corresponds to a natural partial order on bipartitions. Moreover, we prove that the local intersection cohomology of the orbit closures is given by certain bipartition a...

We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron–Garsia nabla operator. The formula involves nested labeled Dyck paths weighted by area and a suitable “diagonal inversion” statistic. Our model includes as special cases many previous conjectures connecting the nabla operator to quantum lattice paths. The combinatorics of the i...

We give a new (inductive) proof of the classical Frobenius– Young correspondence between irreducible complex representations of the symmetric group and Young diagrams, using the new approach, suggested in [11], [15], to determining this correspondence. We also give linear relations between Kostka numbers that follow from the decomposition of the restrictions of induced representations to the pr...

In the large rank limit, for any nonexceptional affine algebra, the graded branching multiplicities known as one-dimensional sums, are conjectured to have a simple relationship with those of type A, which are known as generalized Kostka polynomials. This is called the X = M = K conjecture. It is proved for tensor products of the “symmetric power” Kirillov-Reshetikhin modules for all nonexceptio...

s for Talks Speaker: Nick Loehr (Virginia Tech, USA) (talk describes joint work with Jim Haglund and Mark Haiman) Title: Symmetric and Non-symmetric Macdonald Polynomials Abstract: Macdonald polynomials have played a central role in symmetric function theory ever since their introduction by Ian Macdonald in 1988. The original algebraic definitions of these polynomials are very nonexplicit and d...