On the existence of tableaux with given modular major index
Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 3-21.

We provide simple necessary and sufficient conditions for the existence of a standard Young tableau of a given shape and major index r mod n, for all r. Our result generalizes the r=1 case due essentially to Klyachko [11] and proves a recent conjecture due to Sundaram [32] for the r=0 case. A byproduct of the proof is an asymptotic equidistribution result for “almost all” shapes. The proof uses a representation-theoretic formula involving Ramanujan sums and normalized symmetric group character estimates. Further estimates involving “opposite” hook lengths are given which are well-adapted to classifying which partitions λn have f λ n d for fixed d. We also give a new proof of a generalization of the hook length formula due to Fomin-Lulov [4] for symmetric group characters at rectangles. We conclude with some remarks on unimodality of symmetric group characters.

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Accepted:
Published online:
DOI: 10.5802/alco.4
Classification: 05E10
Keywords: Standard Young tableaux, symmetric group characters, major index, hook length formula, rectangular partitions
Swanson, Joshua P. 1

1 University of Washington Dept. Mathematics Seattle, WA 98195-4350 (USA)
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Swanson, Joshua P. On the existence of tableaux with given modular major index. Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 3-21. doi : 10.5802/alco.4. http://archive.numdam.org/articles/10.5802/alco.4/

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