Skew Howe duality and random rectangular Young tableaux

Abstract

We consider the decomposition into irreducible components of the external power ${\Lambda }^p({\mathbb{C}}^m\otimes {\mathbb{C}}^n)$ regarded as a ${\mathrm{GL}}_m\times {\mathrm{GL}}_n$-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.~$\mu ={\lambda }^{\prime }$. We show that the Young diagram $\lambda$ which corresponds to a randomly selected irreducible component $(\lambda ,{\lambda }^{\prime })$ has the same distribution as the Young diagram which consists of the boxes with entries $\le p$ 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.

17 pages. Version 2: change of title, section on bijective proofs improved