A Note on the Symmetric Powers of the Standard Representation of $S_n$

Abstract

In this paper, we prove that the dimension of the space span- ned by the characters of the symmetric powers of the standard $n$-dimensional representation of $S_n$ is asymptotic to $n^2 / 2$. This is proved by using generating functions to obtain formulas for upper and lower bounds, both asymptotic to $n^2/2$, for this dimension. In particular, for $n \ge 7$, these characters do not span the full space of class functions on $S_n$.