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 p ( m n ) regarded as a GL m ×GL n -module. Skew Howe duality implies that the Young diagrams from each pair (λ,μ) which contributes to this decomposition turn out to be conjugate to each other, i.e. μ=λ . We show that the Young diagram λ which corresponds to a randomly selected irreducible component (λ,λ ) has the same distribution as the Young diagram which consists of the boxes with entries 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 tend to infinity.

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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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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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