On the existence of tableaux with given modular major index

Swanson, Joshua P.

doi:10.5802/alco.4

Swanson, Joshua P.¹

¹ University of Washington Dept. Mathematics Seattle, WA 98195-4350 (USA)

Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 3-21.

Abstract

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 $λ ⊢ n$ have $f^{λ} \leq n^{d}$ 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
Published online: 2018-01-29

MR Zbl

DOI: 10.5802/alco.4

Classification: 05E10
Keywords: Standard Young tableaux, symmetric group characters, major index, hook length formula, rectangular partitions

Author's affiliations:

Swanson, Joshua P. ¹

¹ University of Washington Dept. Mathematics Seattle, WA 98195-4350 (USA)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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. https://alco.centre-mersenne.org/articles/10.5802/alco.4/

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