  We introduce a "second-quantized" representation of the ring of symmetric
functions to further develop a purely second-quantized -- or "lattice" --
approach to the study of zero modes of frustration free
Haldane-pseudo-potential-type Hamiltonians, which in particular stabilize
Laughlin ground states. We present three applications of this formalism. We
start demonstrating how to systematically construct all zero-modes of
Laughlin-type parent Hamiltonians in a framework that is free of
first-quantized polynomial wave functions, and show that they are in one-to-one
correspondence with dominance patterns. The starting point here is the
pseudo-potential Hamiltonian in "lattice form", stripped of all information
about the analytic structure of Landau levels (dynamical momenta). Secondly, as
a by-product, we make contact with the bosonization method, and obtain an
alternative proof for the equivalence between bosonic and fermionic Fock
spaces. Finally, we explicitly derive the second-quantized version of Read's
non-local (string) order parameter for the Laughlin state, extending an earlier
description by Stone. Commutation relations between the local quasi-hole
operator and the local electron operator are generalized to various geometries.
