BPS states, conserved charges and centres of symmetric group algebras

In $\mathcal{N}=4$ SYM with $U(N)$ gauge symmetry, the multiplicity of half-BPS states with fixed dimension can be labelled by Young diagrams and can be distinguished using conserved charges corresponding to Casimirs of $U(N)$. The information theoretic study of LLM geometries and superstars in the dual $AdS_5 \times S^5$ background has raised a number of questions about the distinguishability of Young diagrams when a finite set of Casimirs are known. Using Schur-Weyl duality relations between unitary groups and symmetric groups, these questions translate into structural questions about the centres of symmetric group algebras. We obtain analytic and computational results about these structural properties and related Shannon entropies, and generate associated number sequences. A characterization of Young diagrams in terms of content distribution functions relates these number sequences to diophantine equations. These content distribution functions can be visualized as connected, segmented, open strings in content space.