Denote by $\mathfrak{S}_k$ the symmetric group on $k$ elements. Let $\lambda=(n^2\times n)=(n^2,\ldots,n^2)$ be a rectangular partition and $k=n^3$. Denote by $S_\lambda$ the Specht module corresponding to $\lambda$ and set $V:=S_\lambda^{\otimes 3}=S_\lambda\otimes S_\lambda\otimes S_\lambda$. There is a (unique, up to scalars) nonzero element $v\in V$ with $\sigma\cdot v=v$ for all $\sigma\in\mathfrak{S}_{n^3}$, this is something I can prove. However, I would like to know what that element looks like. Does anyone have any idea?
Invariant element in the tensor product of rectangular Specht modules?
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representation-theory
tensor-products
integer-partitions
symmetric-groups
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0A second approach would be to look at the natural projection operator onto the trivial rep, namely $\displaystyle\operatorname{Pr}(v)=\frac{1}{|G|}\sum_{g\in G} gv$ (which works for any group, hence the more generic notation). Unfortunately, it will likely be nontrivial to show that for a particular $v$ that $\operatorname{Pr}(v)\neq 0$. – 2012-06-06