The -Young lattice is a partial order on partitions with no part larger than . This weak subposet of the Young lattice originated (Duke Math. J. 116 (2003) 103–146) from the study of the -Schur functions , symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed by -bounded partitions. The chains in the -Young lattice are induced by a Pieri-type rule experimentally satisfied by the -Schur functions. Here, using a natural bijection between -bounded partitions and -cores, we establish an algorithm for identifying chains in the -Young lattice with certain tableaux on cores. This algorithm reveals that the -Young lattice is isomorphic to the weak order on the quotient of the affine symmetric group by a maximal parabolic subgroup. From this, the conjectured -Pieri rule implies that the -Kostka matrix connecting the homogeneous basis to may now be obtained by counting appropriate classes of tableaux on -cores. This suggests that the conjecturally positive -Schur expansion coefficients for Macdonald polynomials (reducing to -Kostka polynomials for large ) could be described by a -statistic on these tableaux, or equivalently on reduced words for affine permutations.
Research of L. Lapointe was supported in part by FONDECYT (Chile) Grant #1030114, the Programa Formas Cuadráticas of the Universidad de Talca, and NSERC (Canada) Grant #250904.