Zero-free columns in character tables of symmetric groups
Colin Defant, Sidharth Hariharan, Kenny Lau, Ken Ono
math.CO
Aug 27, 2026 · v1
math.NT math.RT
TL;DR
AxiomProver formalized the paper's results on zero-free character-table columns in Lean, assuming preexisting literature.
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 Matomäki 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.
Problem
Let D(n) count conjugacy classes of the symmetric group S_n whose character-table column contains no zero entry. The goal is to bound D(n), improving on the known D(n) ≪ n^2.
Approach
Duro's reduction restricts zero-free columns to cycle types (3^a,2^b,1^c) with b even. The authors bound the exponents via distances of n to 2-core and 3-core sizes using the Murnaghan–Nakayama rule and a core-removal cutoff lemma. An almost-all refinement uses the Matomäki–Radziwiłł gap theorem applied to a multiplicative set of Loeschian numbers. The results were formalized in Lean by AxiomProver assuming preexisting literature.
Results
They prove D(n) ≪ n^{3/4}, and for almost all n that D(n) ≪_B n^{1/2}(log n)^B for every B>5/6, with a quantitative exceptional set. A heuristic supports the conjecture D(n) ≪_ε n^ε.
