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Tableaux on $$k+1$$-cores, reduced words for affine permutations, and $$k$$-Schur expansions. (English) Zbl 1120.05093
The paper under review focuses on $$k$$ bounded partitions – those whose largest part is less than or equal to $$k$$. These partitions have associated to them a $$k$$-Young lattice and $$k$$-Schur functions, and the related $$k$$-tableaux are defined in terms of $$k+1$$-cores. These objects originated in the study of Macdonald polynomials [L. Lapointe, A. Lascoux and J. Morse, Duke J. Math. 116, 103–146 (2003; Zbl 1020.05069); L. Lapointe and J. Morse, J. Comb. Theory, Ser. A 101, No. 2, 191–224 (2003; Zbl 1018.05101)].
The main result here establishes a bijection between particular saturated chains in the $$k$$-Young lattice and standard $$k$$-tableaux of particular shape. These $$k$$-tableaux are then conjectured to enumerate certain terms in the $$k$$-Schur function expansion of the Macdonald polynomials.

##### MSC:
 500000 Symmetric functions and generalizations 5e+10 Combinatorial aspects of representation theory
##### Keywords:
$$k$$-Schur functions; $$k$$-cores; Macdonald polynomials
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##### References:
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