Decomposition of the symmetric square of the standard representation of the symetric group

Question

As C Monsour already ansered part of my question, I edit it.

The only remaining question is how to decompose into irreducible representations the symmetric square $V$ of the standard representation $\phi$ of the symmetric group? Everything is defined in the following, which I kept inchanged:

I'd like to decompose and find an adapted basis for the decomposition into irreducible of the following representation:

Consider the symetric groupe $S_n$, acting over $\{e_i\}$, the canonical basis of $\Bbb{C}^n$ by permutation of indices. Then the standard representation $\phi$ of $S_n$ is generated by the $\delta^-_{ij}=e_i-e_j$.

I would like to decompose into irreducible the representation $\phi\otimes\phi$. I found a first stable subspace which is I guess irreducible:

• $V$ generated by the $\delta^-_{ij}\otimes\delta^-_{ij}$ is stable, of dimension $n(n-1)/2$.
• Then its orthogonal representation $W$, of dimension $(n-1)(n-2)/2$, is also stable.

I have the following questions:

1. is $V$ irreducible in general?
2. is $W$ irreducible in general?
3. is it possible to easely exibit a basis of $W$?

Thanks a lot!

Answer 1

To answer the revised question, the symmetric square of $\phi$ decomposes as a sum of three irreducible representations: The principal character, the standard representation, and an irreducible representation of dimension $\frac{n(n-3)}{2}$. See the formula near the top of page four in Bowman et al (https://arxiv.org/pdf/1210.5579.pdf), where the other representation in the sum is the alternating square.

Answer 2

I believe your notation is confusing. Let $\zeta_i=e_{1+1}-e_i$ for $1\le i \le n-1$. Then the $\zeta_i$ are a linear basis for $\phi$, the $\zeta_i \otimes \zeta_j + \zeta_j \otimes \zeta_i$ for $1\le i\le j\le n-1$ are a basis for $V$ and the $\zeta_i \otimes \zeta_j - \zeta_j \otimes \zeta_i$, for $1\le i<j\le n-1$ are a basis of W. If that is what you intended, that also answers (c).

In his book Linear Representations of Finite Groups, Serre calls $V$ and $W$ the symmetric and alternating squares of $\phi$.

$\phi$ is codimension 1 in the natural permutation representation of $S_n$. It is irreducible based on standard results about multiply transitive permutation groups. (See, e.g., Representation of multiply transitive group .)

(a): In general $V$ cannot be irreducible. For example, for $S_5$, $\dim V=10$ but no irreducible representation has dimension larger than $6$, as you can tell by looking up or constructing the character table.

(b): While alternating squares of irreducibles are not in general irreducible (if the highest dimension irreducible has dimension at least $5$ then its alternating square is never irreducible, for example), I'm fairly certain the alternating square of $\phi$ is irreducible. I don't recall the proof, however. There is a lot of literature on the representation theory of the symmetric group, on which I am not an expert. I am sure you can find a detailed answer there. But I suspect you can find a proof that just relies on the multiple transitivity of $S_n$.

(c) See above.

Answer 3

Ok thanks to C Monsour I found the answer.

First look at https://arxiv.org/pdf/1210.5579.pdf page 4. The correspondance in notations are $\phi:=S(n-1,1), T:=S(n), \psi=S(n-2,1^2), W=S(n-2,1^2)$.

It is said that $\phi\otimes\phi=T\oplus\phi\oplus\psi\oplus W$. We already have a basis for $W$ and for $V:=T\oplus\phi\oplus\psi$.

In the following I give explicit correspondence with the usual way of presenting $T$ and $\phi$. As $\psi$ is the orthogonal space of $T\oplus\phi$, it is sufficient.

A basis of $V$ is given by the $\{\alpha_{ij}\}_{i<j}$ where $\alpha_{ij}:=\delta_{ij}^-\otimes \delta_{ij}^-$ (remark that $\alpha_{ij}=\alpha_{ji}$). Let $$t=\sum_{i<j}\alpha_{ij}$$ and $$e_k=\sum_{i\neq k} \alpha_{ik}-\sum_{i,j\neq k;i\neq j} \alpha_{ij}+\lambda t,$$ where $\lambda$ is chosen such that $t$ and $e_k$ are orthogonals.

Then, $t$ clearly generates the trivial irrep $T$. Moreover, the e_k generates a space of dimension $n-1$ and a permutation $\sigma$ applied to $e_k$ gives $\sigma(e_k)=e_{\sigma(k)}$. This space is $\phi$ and the $e_k$ are the vertex of a regular polytope in dimension $n$.