A Note on the Symmetric Powers of the Standard Representation of Sn
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 Sn is asymptotic to n2/2. This is proved by using generating functions to obtain formulas for upper and lower bounds, both asymptotic to n2/2, for this dimension. In particular, for n>6, these characters do not span the full space of class functions on Sn.
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