Fluctuations for standard Young tableaux

We will introduce the Young generating function and use it to characterize the law of large numbers and the central limit theorem behaviors for random partitions. As an application of these results, we present a framework to obtain conditional Gaussian Free Field fluctuations for height functions associated with random standard Young tableaux. To prove these results we develop algebraic formulas for operators on the Gelfand–Tsetlin algebra of the symmetric group.