Geodesic
A note on symmetric powers of the standard representation of \(S_n\)
The electronic journal of combinatorics, Tome 7 (2000)
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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 $S_n$.
DOI : 10.37236/1484
Classification : 05E10, 05A15, 05A16, 05E05
Mots-clés : standard representation, characters 
@article{10_37236_1484,
     author = {D. Savitt and R. P. Stanley},
     title = {A note on symmetric powers of the standard representation of {\(S_n\)}},
     journal = {The electronic journal of combinatorics},
     year = {2000},
     volume = {7},
     doi = {10.37236/1484},
     zbl = {0959.05119},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1484/}
}
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D. Savitt; R. P. Stanley. A note on symmetric powers of the standard representation of \(S_n\). The electronic journal of combinatorics, Tome 7 (2000). doi: 10.37236/1484

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