Relations between values and zeros of irreducible characters of symmetric groups

TL;DR

This paper establishes polynomial relations among irreducible character values of symmetric groups, explores zeros of characters, and develops algorithms for character value computation, advancing understanding of character theory in symmetric groups.

Contribution

It introduces new polynomial relations, characterizes zeros of irreducible characters, and provides recursive algorithms for character value calculations in symmetric groups.

Findings

Certain conjugacy classes cannot be simultaneously zeros for all irreducible characters.

Values of 2-defect zero characters are rational functions in n.

Improved methods for identifying irreducible characters using fewer character values.

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.