Skew Howe duality and random rectangular Young tableaux

Panova, Greta; Śniady, Piotr

Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 81-94.

We consider the decomposition into irreducible components of the external power ${\bigwedge }^p({ℂ}^m\otimes {ℂ}^n)$ regarded as a ${GL}_m\times {GL}_n$-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.

Classification: 22E46, 20C30, 60C05
Keywords: Skew Howe duality, random Young diagrams, representations of general linear groups $\protect \operatorname{GL}_m$, representations of finite symmetric groups

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