# ALGEBRAIC COMBINATORICS

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 $\lambda ⊢n$ have ${f}^{\lambda }\le {n}^{d}$ 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.
Accepted : 2017-08-19
Published online : 2018-01-29
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},
zbl = {06882332},
mrnumber = {3857157},
language = {en},
url = {https://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, pp. 3-21. doi : 10.5802/alco.4. https://alco.centre-mersenne.org/item/ALCO_2018__1_1_3_0/

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