Filling the Bose sea: symmetric quantum Hall edge states and affine characters

We explore the structure of the bosonic analogues of the $k$-clustered ``parafermion'' quantum Hall states. We show how the many-boson wave functions of $k$-clustered quantum Hall droplets appear naturally as matrix elements of ladder operators in integrable representations of the affine Lie algebra $\hat{su}(2)_k$. Using results of Feigin and Stoyanovsky, we count the dimensions of spaces of symmetric polynomials with given $k$-clustering properties and show that as the droplet size grows the partition function of its edge excitations evolves into the character of the representation. This confirms that the Hilbert space of edge states coincides with the representation space of the $\hat{su}(2)_k$ edge-current algebra. We also show that a spin-singlet, two-component $k$-clustered boson fluid is similarly related to integrable representations of $\hat{su}(3)$. Parafermions are not necessary for these constructions.