# ALGEBRAIC COMBINATORICS

Skew Howe duality and random rectangular Young tableaux
Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 81-94.

We consider the decomposition into irreducible components of the external power ${\bigwedge }^{p}\left({ℂ}^{m}\otimes {ℂ}^{n}\right)$ regarded as a ${GL}_{m}×{GL}_{n}$-module. Skew Howe duality implies that the Young diagrams from each pair $\left(\lambda ,\mu \right)$ 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 $\left(\lambda ,{\lambda }^{\prime }\right)$ 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.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.8
Classification: 22E46,  20C30,  60C05
Keywords: Skew Howe duality, random Young diagrams, representations of general linear groups ${GL}_{m}$, representations of finite symmetric groups
Panova, Greta 1; Śniady, Piotr 2

1 UPenn Mathematics Department, 209 South 33rd St, Philadelphia, PA 19104, USA
2 Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland
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Panova, Greta; Śniady, Piotr. Skew Howe duality and random rectangular Young tableaux. Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 81-94. doi : 10.5802/alco.8. https://alco.centre-mersenne.org/articles/10.5802/alco.8/

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