We provide simple necessary and sufficient conditions for the existence of a standard Young tableau of a given shape and major index mod , for all . Our result generalizes the case due essentially to Klyachko [11] and proves a recent conjecture due to Sundaram [32] for the 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 have for fixed . 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.
Accepted:
Published online:
DOI: 10.5802/alco.4
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}, pages = {3--21}, publisher = {MathOA foundation}, volume = {1}, number = {1}, year = {2018}, doi = {10.5802/alco.4}, zbl = {06882332}, mrnumber = {3857157}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/alco.4/} }
TY - JOUR AU - Swanson, Joshua P. TI - On the existence of tableaux with given modular major index JO - Algebraic Combinatorics PY - 2018 SP - 3 EP - 21 VL - 1 IS - 1 PB - MathOA foundation UR - http://archive.numdam.org/articles/10.5802/alco.4/ DO - 10.5802/alco.4 LA - en ID - ALCO_2018__1_1_3_0 ER -
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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