Young supertableaux and the large $\mathcal{N}=4$ superconformal algebra

In this paper we consider unitary highest weight irreducible representations of the `Large’ $\mathcal{N}=4$ superconformal algebra $A_{\gamma }$ in the Ramond sector as infinite-dimensional graded modules of its zero mode subalgebra, $\mathfrak{s}\mathfrak{u}(2\mathrm{\mid }2)$. We describe how representations of $\mathfrak{s}\mathfrak{u}(2\mathrm{\mid }2)$ may be classified using Young supertableaux, and use the decomposition of $A_{\gamma }$ as an $\mathfrak{s}\mathfrak{u}(2\mathrm{\mid }2)$ module to discuss the states which contribute to the supersymmetric index $I_1$, previously proposed in the literature by Gukov, Martinec, Moore and Strominger.