Relations between values and zeros of irreducible characters of symmetric groups
Lee Tae Young
TL;DR
This work studies polynomial relations among the values of irreducible characters of symmetric groups $S_n$ to understand zeros and character identification. It proves the existence of universal polynomials $T_\lambda$ that express normalized character values $\rho(\lambda)$ in terms of values on smaller cycle-types, and provides a Gröbner-basis framework to deduce zero-distribution constraints, with explicit $T_\lambda$ for small norms. It then analyzes zeros of irreducible characters, deriving numerous explicit forbidden zero-sets and applying Siegel’s theorem to bound integral points, yielding finite classifications in many cases. The paper also develops a polynomial framework for $2$-defect zero (staircase) characters and extends to $3$-defect zero cases, linking to generalized octagonal numbers and exploring implications for tensor-square conjectures. Finally, it argues that, beyond the established $T_\lambda$-driven relations, there is no broad polynomial relation among $n$ and finite sets of character-values that holds uniformly across all large $n$, highlighting intrinsic limits to such realizations.
Abstract
We prove certain polynomial relations between the values of complex irreducible characters of general finite symmetric groups. We use it to find some sets of conjugacy classes such that no finite symmetric group has a complex irreducible character that vanishes at every class in the set. In particular, we show that if $n$ satisfies certain conditions, then $S_n\setminus \{1\}$ cannot be covered by the set of zeros of three irreducible characters. We also prove that the values of character of $2$-defect zero can be expressed as rational functions in $n$, and build a recursive algorithm to find these rational functions. As another application, we improve a result by A. Miller on identification of irreducible characters by checking small number of values.