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Skeleton diagrams are usually used to discuss general properties of the perturbation series in field theory. They help to prove renormalizability of a theory, or to prove properties of correlation functions. However, they are not used in general for explicit calculation. (The main counter example is the Diagramatic Monte Carlo, that tries to compute high ...

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You may find the following paper useful: A Symbolic Solution of the Hubbard Model for Small Clusters, by J. Yepez. You may also want to review group theory for condensed matter physics, because your questions essentially span the basics of group and representation theory. Many texts give good overviews of the fundamentals of group theory as applied to ...

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No doubt that a must read on this topic is the classic work by Fradkin and Shenker: http://journals.aps.org/prd/abstract/10.1103/PhysRevD.19.3682 In particular, it was pointed out that for $Z_2$ gauge theories (and I believe for all $Z_n$) the confined phase and the Higgs phase are in fact smoothly connected. There is no sharp phase boundary between the ...

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