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We introduce a "second-quantized" representation of the ring of symmetricfunctions to further develop a purely second-quantized -- or "lattice" --approach to the study of zero modes of frustration freeHaldane-pseudo-potential-type Hamiltonians, which in particular stabilizeLaughlin ground states. We present three applications of this formalism. Westart demonstrating how to systematically construct all zero-modes ofLaughlin-type parent Hamiltonians in a framework that is free offirst-quantized polynomial wave functions, and show that they are in one-to-onecorrespondence with dominance patterns. The starting point here is thepseudo-potential Hamiltonian in "lattice form", stripped of all informationabout the analytic structure of Landau levels (dynamical momenta). Secondly, asa by-product, we make contact with the bosonization method, and obtain analternative proof for the equivalence between bosonic and fermionic Fockspaces. Finally, we explicitly derive the second-quantized version of Read'snon-local (string) order parameter for the Laughlin state, extending an earlierdescription by Stone. Commutation relations between the local quasi-holeoperator and the local electron operator are generalized to various geometries.

Zero modes, Bosonization and Topological Quantum Order: The Laughlin State in Second Quantization