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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.

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