  We consider the decomposition into irreducible components of the external
power $\Lambda^p(\mathbb{C}^m\otimes \mathbb{C}^n)$ regarded as a
$\operatorname{GL}_m\times\operatorname{GL}_n$-module. Skew Howe duality
implies that the Young diagrams from each pair $(\lambda,\mu)$ which
contributes to this decomposition turn out to be conjugate to each other,
i.e.~$\mu=\lambda'$. We show that the Young diagram $\lambda$ which corresponds
to a randomly selected irreducible component $(\lambda,\lambda')$ has the same
distribution as the Young diagram which consists of the boxes with entries
$\leq p$ of a random Young tableau of rectangular shape with $m$ rows and $n$
columns. This observation allows treatment of the asymptotic version of this
decomposition in the limit as $m,n,p\to\infty$ tend to infinity.
