Orthogonal Rotation Group

Orthogonal rotation groups are Lie Groups. The orthogonal rotation group $O_3(n)$ is the set of $n\times n$ Real Orthogonal Matrices.

The orthogonal rotation group $O_3^-(n)$ is the set of $n\times n$ Real Orthogonal Matrices (having $n(n-1)/2$ independent parameters) with Determinant $-1$.

The orthogonal rotation group $O_3^+(n)$ is the set of $n\times n$ Real Orthogonal Matrices, having $n(n-1)/2$ independent parameters, with Determinant $+1$. $O_3^+(n)$ is Homeomorphic with ${\it SU}(2)$. Its elements can be written using Euler Angles and Rotation Matrices as

$$I = \left[{\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}}\right]$$ (1)

$$R_x(\phi) = \left[{\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos\phi & \sin\phi \\ 0 & -\sin\phi & \cos\phi\end{array}}\right]$$ (2)

$$R_y(\theta) = \left[{\begin{array}{ccc}\cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \\ \sin\theta & 0 & \cos\theta\end{array}}\right]$$ (3)

$$R_z(\psi) = \left[{\begin{array}{ccc}\cos\psi & \sin\psi & 0 \\ -\sin\psi & \cos\psi & 0\\ 0 & 0 & 1\end{array}}\right].$$ (4)

References

Arfken, G. ``Orthogonal Group, $O_3^+$.'' Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 252-253, 1985.

Wilson, R. A. ``ATLAS of Finite Group Representation.'' http://for.mat.bham.ac.uk/atlas/html/contents.html#orth.