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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.

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    $\begingroup$ Do you mean $U$ is the irreducible representation corresponding to the partition $(n-1,1)$ or do you mean the reducible representation $\mathbb{C}^n$ with the permutation of coordinates? $\endgroup$
    – Abdelmalek Abdesselam
    Commented Apr 10, 2019 at 16:33
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    $\begingroup$ @AbdelmalekAbdesselam I believe the permutation representation refers to the reducible one, while the irreducible one is the standard representation. $\endgroup$
    – Mike Earnest
    Commented Apr 10, 2019 at 17:35

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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.

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