Tableaux on -cores, reduced words for affine permutations, and k-Schur expansions

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 (Duke Math. J. 116 (2003) 103–146) from the study of the k -Schur functions s λ ( k ) , 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 satisfied by the k -Schur functions. Here, using a natural bijection between k -bounded partitions and k + 1 -cores, we establish an algorithm for identifying chains in the k -Young lattice with certain tableaux on k + 1 cores. This algorithm reveals that the k -Young lattice is isomorphic to the weak order on the quotient of the affine symmetric group S ˜ k + 1 by a maximal parabolic subgroup. From this, the conjectured k -Pieri rule implies that the k -Kostka matrix connecting the homogeneous basis { h λ } λ ∈ Y k to { s λ ( k ) } λ ∈ Y k may now be obtained by counting appropriate classes of tableaux on k + 1 -cores. This suggests that the conjecturally positive k -Schur expansion coefficients for Macdonald polynomials (reducing to q , t -Kostka polynomials for large k ) could be described by a q , t -statistic on these tableaux, or equivalently on reduced words for affine permutations.