A group method solving many-body systems in intermediate statistical representation

The exact solution of the interacting many-body system is important and difficult to solve. In this paper, we introduce a group method solve problem using relation between permutation unitary group. We prove theorem first, then theorem, represent Hamiltonian by Casimir operators eigenvalues could give values energy thus those problems exactly. This maps onto an intermediate statistical representation. conjugacy-class operator in representation, called Gentile Bose Fermi cases are two limitations also discuss representation space symmetric example Heisenberg model demonstrate method. It shown that effective problems.