Limit shape for infinite rank limit of tensor power decomposition for Lie algebras of series so(2n+1)

Abstract

We consider the Plancherel measure on irreducible components of tensor powers of the spinor representation of so(2n+1). The irreducible representations correspond to the generalized Young diagrams. With respect to this measure the probability of an irreducible representation is the product of its multiplicity and dimension, divided by the total dimension of the tensor product. We study the limit shape of the generalized Young diagram when the tensor power N and the rank n of the algebra tend to infinity with N/n fixed. We derive an explicit formula for the limit shape and prove convergence to it in probability. We prove central limit theorem for global fluctuations around the limit shape.

36 pages, 7 figures. In version4 we have added proof of central limit theorem for global fluctuations around the limit shape, that relies on Christoffel transformation of Krawtchouk orthogonal polynomials. Discussion of relation to Berele insertion and skew Howe duality is added. The code that was used to produce the Figures is available at https://github.com/naa/young-diagrams/