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Skew Howe duality and random rectangular Young tableaux. (English) Zbl 1391.05268
Summary: We consider the decomposition into irreducible components of the external power $$\bigwedge^p(\mathbb{C}^m\otimes \mathbb{C}^n)$$ 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 $$m,n,p\rightarrow \infty$$ tend to infinity.

##### MSC:
 05E10 Combinatorial aspects of representation theory 22E46 Semisimple Lie groups and their representations 20C30 Representations of finite symmetric groups 60C05 Combinatorial probability
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