Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions

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dc.creatorLapointe, L.
dc.creatorMorse, J.
dc.date2004-02-19
dc.date.accessioned2026-07-07T05:05:36Z
dc.date.available2026-07-07T05:05:36Z
dc.descriptionThe $k$-Young lattice $Y^k$ is a partial order on partitions with no part larger than $k$. This weak subposet of the Young lattice originated from the study of the $k$-Schur functions(atoms) $s_λ^{(k)}$, 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 $\tilde S_{k+1}$ by a maximal parabolic subgroup. From this, the conjectured $k$-Pieri rule implies that the $k$-Kostka matrix connecting the homogeneous basis $\{h_\la\}_{\la\in\CY^k}$ to $\{s_\la^{(k)}\}_{\la\in\CY^k}$ 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.
dc.description30 pages, 1 figure
dc.identifierhttps://arxiv.org/abs/math/0402320
dc.identifierhttp://arxiv.org/abs/math/0402320
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/70224
dc.subjectCombinatorics
dc.subjectQuantum Algebra
dc.subject05
dc.titleTableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions
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