College of Arts and Sciences Tableaux on K + 1-cores, Reduced Words for Affine Permutations, and K-schur Expansions

The following item is made available as a courtesy to scholars by the author(s) and Drexel University Library and may contain materials and content, including computer code and tags, artwork, text, graphics, images, and illustrations (Material) which may be protected by copyright law. Unless otherwise noted, the Material is made available for non profit and educational purposes, such as research, teaching and private study. For these limited purposes, you may reproduce (print, download or make copies) the Material without prior permission. All copies must include any copyright notice originally included with the Material. You must seek permission from the authors or copyright owners for all uses that are not allowed by fair use and other provisions of the U.S. Copyright Law. The responsibility for making an independent legal assessment and securing any necessary permission rests with persons desiring to reproduce or use the Material. Abstract. 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 [9] from the study of the k-Schur functions s (k) λ , 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 (k) λ } 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.