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Tableaux on k + 1-cores, reduced words for affine permutations, and k-Schur expansions
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http://hdl.handle.net/1860/1936
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| Title: | Tableaux on k + 1-cores, reduced words for affine permutations, and k-Schur expansions |
| Authors: | Morse, Jennifer
Lapointe, Luc |
| Issue Date: | 19-Feb-2004 |
| Citation: | Retrieved October 30, 2007 from http://lanl.arxiv.org/find/math/1/au:+Morse_J/0/1/0/all/0/1 |
| 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)
,
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 } 2Y k to {s
(k)
} 2Y 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. |
| URI: | http://lanl.arxiv.org/abs/math/0402320v1
http://hdl.handle.net/1860/1936 |
| Appears in Collections: | Faculty Research and Publications (Mathematics)
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