Anton Nazarov, Olga Postnova, Travis Scrimshaw, Skew Howe duality and limit shapes of Young diagrams

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We consider the skew Howe duality for the action of certain dual pairs of Lie groups on the exterior algebra of the product of fundamental representations as a probability measure on Young diagrams by the decomposition into the sum of irreducible representations, in particular, for the exterior algebra of the fundamental representation of the GL series and for the spinor representation in the SO series. In the limit of infinite group rank and infinite tensor power, we have obtained the limit shape of Young diagrams, which parametrize the irreducible representations included in the decomposition. We show that the limit shape is described by one analytical formula  for all classical series. Uniform convergence to the limit shape is proved. It is shown that the limit shape and fluctuations around it are described by the Krawtchouk ensemble.

https://arxiv.org/abs/2111.12426

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