TITLE: Decomposition of tensor product of permutation representation of $S_n$.
QUESTION [5 upvotes]: Please be kind to me - I'm a combinatorist so this question might be a bit naive...
If $U$ is the representation space of the permutation representation of $S_n$, is there any known decomposition into irreducibles of $U\otimes U$?
Cheers.

REPLY [3 votes]: Let $V_{\lambda}$ be the irreducible representation corresponding to the integer partition $\lambda=(\lambda_1,\lambda_2,\ldots)$ of $n$. Then, as explained by Mike Earnest
$$
U=V_{(n)}\oplus V_{(n-1,1)}\ .
$$
since tensoring with the trivial representation $V_{(n)}$ does nothing, it's just a matter of computing some simple Kronecker coefficients which gives
$$
V_{(n-1,1)}\otimes V_{(n-1,1)}=
V_{(n)}\oplus V_{(n-1,1)}\oplus V_{(n-2,2)}\oplus V_{(n-2,1,1)}
$$
with the end result
$$
U\otimes U=2V_{(n)}\oplus 3 V_{(n-1,1)}\oplus V_{(n-2,2)}\oplus V_{(n-2,1,1)}
$$
where $2V_{(n)}$ means the module appreas with multiplicity 2, etc.
If interested in the explicit morphisms, then see this article by Chipalkatti and Mohammed.