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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:
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
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