Polynomial Expressions for Symmetric Group Characters on Cycles

TL;DR

This paper generalizes Cohen and Zemel’s polynomial formulas for symmetric group representation dimensions to character values on arbitrary cycles, revealing new polynomial expressions and combinatorial interpretations.

Contribution

It extends existing polynomial formulas from dimensions to character values on cycles, providing a broader understanding of symmetric group characters.

Findings

Polynomial expressions for symmetric group characters on cycles

Coefficients count specific standard Young tableaux

Generalization of previous dimension formulas

Abstract

In \cite{[CZ]}, Cohen and Zemel showed that for a partition $\lambda \vdash k$, the dimension of the irreducible representation of $S_{n}$ corresponding to the partition $(n-k,\lambda) \vdash n$ is a polynomial of degree $k$ in $n$, whose coefficients in the binomial basis count standard Young tableaux of shape $\lambda$ with special restrictions. In this paper, we generalize their results on the representation's dimension to character values on arbitrary cycles.