Skew Howe duality and random rectangular Young tableaux

Abstract

We consider the decomposition into irreducible components of the external power $Λ^p(\mathbb{C}^m\otimes \mathbb{C}^n)$ regarded as a $\operatorname{GL}_m\times\operatorname{GL}_n$-module. Skew Howe duality implies that the Young diagrams from each pair $(λ,μ)$ which contributes to this decomposition turn out to be conjugate to each other, i.e.~$μ=λ'$. We show that the Young diagram $λ$ which corresponds to a randomly selected irreducible component $(λ,λ')$ 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.