On some classical vector models in statistical mechanics

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Abstract
The partition functions and the two-body correlation functions are obtained for a class of one-dimensional models. Extensive use of the theory of group representations makes is possible to determine straightforwardly the spectral decomposition for the transfer operator. Explicit calculations are performed for some simple models with and without external field. In particular a lattice version for a nematic liquid crystal is obtained. One also shows how the formalism introduced can be used to study the influence of boundary conditions. One finally discusses some possible generalizations to higher-dimensional systems.