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

TL;DR

This paper investigates the asymptotic behavior of generalized Young diagrams associated with tensor powers of the spinor representation of so(2n+1), deriving an explicit limit shape and analyzing fluctuations as parameters grow large.

Contribution

It provides the first explicit formula for the limit shape of these diagrams and establishes convergence and fluctuation results in the asymptotic regime.

Findings

Explicit formula for the limit shape of generalized Young diagrams.

Proved convergence to the limit shape in probability.

Established a central limit theorem for fluctuations around the limit shape.

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.