Mathematical Excursion. Group Characters

Abstract

In this chapter a special group theoretical concept is introduced which has many applications. It describes the main properties of representations and is therefore called “group character”. It solves the problem of how to describe the invariant properties of a group representation in a simple way. If we denote an element of a group G by Ĝ_a, a representation $% feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn \hat D({\hat G_a})$ is not unambiguous, because every similarity transformation $\\hat A\hat D({\hat G_a}){\hat A^{ - 1}},\hat A \in D(G)$ yields an equivalent form. One possibility for the description of the invariant properties would be to use the eigenvalues of the representation matrix, which do not change under a similarity transformation. This leads to the construction of the Casimir operators, the eigenvalues of which classify the representations. The construction of the Casimir operators and their eigenvalues is in general a very difficult nonlinear problem. Fortunately, in many cases it is sufficient to use a simpler invariant, namely the trace of the representation matrix

$$\chi \left( {{{\hat G}_a}} \right) = \sum\limits_{i = 1}^d {{D_{ii}}} \left( {{{\hat G}_a}} \right)$$

(1)

where d is the dimension of the matrix representation. Equation (10.1) is in fact invariant under similarity transformations, because

$$\begin{array}{l}\psi - (\hat G_a ) = \sum\limits_i {D\prime_{ii} (\hat G_a ) = \sum\limits_{ijk} {A_{ij} D_{jk} (\hat G_a )(\hat A^{ - 1} )_{ki} } } \\= \sum\limits_{jk} {D_{jk} (\hat A^{ - 1} \hat A)_{kj} } = \sum\limits_j {D_{jj} (\hat G_a )} = \chi (\hat G_a ){\rm{(10}}{\rm{.2)}} \\\end{array}$$

(10.2)

χ(G _a ) is called the “group character” of the representation.