On the existence of tableaux with given modular major index
Algebraic Combinatorics, Volume 1 (2018) no. 1, p. 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.
Received : 2017-08-19
Accepted : 2017-08-19
DOI : https://doi.org/10.5802/alco.4
Classification:  05E10
Keywords: Standard Young tableaux, symmetric group characters, major index, hook length formula, rectangular partitions
@article{ALCO_2018__1_1_3_0,
     author = {Swanson, Joshua P.},
     title = {On the existence of tableaux with given modular major index},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {1},
     number = {1},
     year = {2018},
     pages = {3-21},
     doi = {10.5802/alco.4},
     language = {en},
     url = {http://alco.centre-mersenne.org/item/ALCO_2018__1_1_3_0}
}
Swanson, Joshua P.. On the existence of tableaux with given modular major index. Algebraic Combinatorics, Volume 1 (2018) no. 1, p. 3-21. doi : 10.5802/alco.4. http://alco.centre-mersenne.org/item/ALCO_2018__1_1_3_0/

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