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

TL;DR

This paper investigates the growth of the space spanned by characters of symmetric powers of the standard representation of S_n, showing it asymptotically approaches n^2/2 and does not cover all class functions for large n.

Contribution

It provides the first asymptotic analysis of the dimension of the character space generated by symmetric powers of the standard representation of S_n.

Findings

Dimension asymptotic to n^2/2 as n grows large

Characters of symmetric powers do not span all class functions for n > 6

Generated bounds are tight and asymptotic to n^2/2

Abstract

In this paper, we prove that the dimension of the space spanned by the characters of the symmetric powers of the standard n-dimensional representation of the symmetric group 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>6, these characters do not span the full space of class functions on S_n.