2004/02/19 by Luc Lapointe, L. Lapointe, Jennifer Morse +3 · 1 citation
Mathematics · #05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA) #math.CO #math.QA #msc:05
paper · pdf · doi:10.48550/arxiv.math/0402320
30 pages, 1 figure
arxiv created 2004/02/19 · openalex publication_date 2004/02/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The k-Young lattice Yk is a partial order on partitions with no part larger than k. This weak subposet of the Young lattice originated from the study of the k-Schur functions(atoms) sλ(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 Sk+1 by a maximal parabolic subgroup. From this, the conjectured k-Pieri rule implies that the k-Kostka matrix connecting the homogeneous basis \h_\la\\la∈\CYk to \s_\la(k)\\la∈\CYk 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.