A prominent class of model FQH ground states is those realized as correlation
functions of $\mathbb{Z}_k^{(r)}$-algebras. In this paper, we study the
interplay between these algebras and their corresponding wavefunctions. In the
hopes of realizing these wavefunctions as a unique densest zero energy state,
we propose a generalization for the projection Hamiltonians. Finally, using
techniques from invariants of binary forms, an ansatz for computation of
correlations $\langle\psi(z_1)\cdots\psi(z_{2k})\rangle
\prod_{i<j}(z_i-z_j)^{2r/k}$ is devised. We provide some evidence that, at
least when $r=2$, our proposed Hamiltonian realizes
$\mathbb{Z}_k^{(2)}$-wavefunctions as a unique ground state.

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