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arXiv · 2608.27718

Zero-free columns in character tables of symmetric groups

Colin Defant·Sidharth Hariharan·Kenny Lau·Ken Ono
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Abstract

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 $D(n)\geq 1$. It is known that $D(n)\ll n^2$. We prove that $D(n)\ll n^{3/4}$. Second, we prove for almost all positive integers $n$ that $D(n)\ll_B n^{1/2}(\log n)^B$ for every $B>5/6$, with a quantitative bound for the exceptional set, using work of Matom\"aki and Radziwill. Finally, we offer a heuristic supporting our conjecture that $D(n)\ll_{\varepsilon} n^{\varepsilon}$. AxiomProver formalized the results in this paper in Lean assuming preexisting literature.

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Colin Defant, Sidharth Hariharan, Kenny Lau, Ken Ono. 2026-08-27. Zero-free columns in character tables of symmetric groups. https://arxiv.org/abs/2608.27718

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