A Study Of Irreducible Tensor Basis Method In Group Representation Theory

The representation theory of simple Lie groups has now been established as one of the most useful tools in modern physics.In this paper, the irreducible tensor basis method in group representation theory is studied further. The main work is:1. The irreducible tensor basis of infinitesimal generators of five exceptional Lie groups G₂, F₄, E₆, E₇ and E₈ are constructured, respectively. Similar to the classical Lie groups, the irreducible tensor basis of five exceptional Lie groups are made up of some independent angular momentum operators and some corrrespond-ing many-fold irreducible tensor operators, which have distinct physics significance. Hence, The kind of generators of exceptional Lie groups may be understood more easy, and the commutation relations satisfied by the kind of generators are more simple and regular. In terms of the irreducible tensor basis, we may solve the problems of irreducible representation of five exceptional Lie groups by using the theory of angular momentum and the Wigner-Eckart theorem, and discussed their applications in the physics and chemistry.2. Irreducible representations and reduction factors of O(N) group are obtained by using the irreducible tensor basis method. We considered the second realization of irreducible tensor basis of 0(N) — the generators of 0{N) include the generators of its subgroup O(N — 1) and a set of 1 rank irreducible tensor operators of 0{N — 1), and constructed its irreducible tensor basis for each subgroup in the canonical group chain 0(N) D 0(N - 1) O(3). We obtained finally, by mathematical induction, the irreducible representations [m_1N m_2N … m_[N/2]N)...