Filling the Bose sea: clustered states and excitations

We explore the structure of clustered quantum states, which might be realized in `droplets' of rapidly rotating Bose Einstein condensates. We explore the underlying algebraic structure (which is that of the affine Lie algebra $su(2)_k$) and count the dimension of the space of symmetric polynomials which have the clustering property. Upon increasing the size of the droplet, the partition function of the droplet becomes a character of the underlying algebra $su(2)_k$, confirming that the system can be described by an $su(2)$ Chern-Simons theory.