Continuous Groups With Representations

• Albert Fässler
• Eduard Stiefel

Abstract

We resume Example 1.5 where the group G of all proper rotations of the space R3 was introduced. After choosing a cartesian coordinate system (i.e., an orthonormal basis) in R3, we can describe such a rotation by the following matrix A: the kth basis vector is rotated into a vector a k with coordinates a1k, a2k, a3k Thus these coordinates must be written in the kth. column of A, thus
$$A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}{a_{12}}{a_{13}}} \\ {{a_{21}}{a_{22}}{a_{23}}} \\ {{a_{31}}{a_{32}}{a_{33}}} \end{array}} \right]$$
The three vectors a k are orthogonal unit vectors. Therefore, the inner product of two distinct columns = 0, and the inner product of a column with itself = 1. Inshort,
$${A^{\text{T}}}A = E{\text{ or equivalently }}{A^{ - 1}} = {A^T}$$
(4.1)
Here A T is the transposed matix and E the 3×3 identity matrix. Real matrices with the property (4.1) are called orthogonal. Conxequently, the inverse of an orthogonal matrix is obtained by interchanging rown and columns.

Keywords

Irreducible Representation Homogeneous Polynomial Lorentz Group Abelian Subgroup Spherical Function
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