Skew Howe duality and random rectangular Young tableaux

Source abstract

We consider the decomposition into irreducible components of the external power ⋀ p ( ℂ m ⊗ ℂ 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.