PreprintCC0arXiv
Zero-free columns in character tables of symmetric groups
Abstract
The rows and columns of the character table of the symmetric group Sn are both naturally indexed by partitions of n. Let D(n) denote the number of conjugacy classes of Sn whose column contains no zero entry. The identity column is always zero-free, so D(n)≥ 1. It is known that D(n)≪ n2. We prove that D(n)≪ n3/4. Second, we prove for almost all positive integers n that D(n)≪B n1/2(log n)B for every B>5/6, with a quantitative bound for the exceptional set, using work of Matomäki and Radziwill. Finally, we offer a heuristic supporting our conjecture that D(n)≪ε nε. AxiomProver formalized the results in this paper in Lean assuming preexisting literature.
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BibTeX
@misc{defant2026zerofree,
title = {{Zero-free columns in character tables of symmetric groups}},
author = {Colin Defant and Sidharth Hariharan and Kenny Lau and Ken Ono},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27718},
eprint = {2608.27718},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27718},
note = {CC0},
}
RIS
TY - UNPB
TI - Zero-free columns in character tables of symmetric groups
AU - Colin Defant
AU - Sidharth Hariharan
AU - Kenny Lau
AU - Ken Ono
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27718
UR - https://arxiv.org/abs/2608.27718
AB - The rows and columns of the character table of the symmetric group S_n are both naturally indexed by partitions of n. Let D(n) denote the number of conjugacy classes of S_n whose column contains no zero entry. The identity column is always zero-free, so D(n)≥ 1. It is known that D(n)≪ n^2. We prove that D(n)≪ n^3/4. Second, we prove for almost all positive integers n that D(n)≪_B n^1/2(log n)^B for every B>5/6, with a quantitative bound for the exceptional set, using work of Matomäki and Radziwill. Finally, we offer a heuristic supporting our conjecture that D(n)≪_ε n^ε. AxiomProver formalized the results in this paper in Lean assuming preexisting literature.
ER -