The Distribution Density of the Commutant of Random Rotations of a Three-Dimensional Euclidean Space

Abstract

The basic measure $\mu $ is defined on the group $ SO (3) $ of rotations of three-dimensional Euclidean space. It responds to the product of uniform distributions on the sets of axes of rotations and angles of rotations. We consider three distribution densities with respect to $\mu$: $\rho_0 $ is a density of left- and right-invariant measure (Haar measure); $ \rho_1 $ is a density of distribution of rotations $ \Lambda^k$, $k \ge 2 $, where $ \Lambda $ is a random rotation with density $ \rho_0 $; and $ \rho_2 $ is a distribution density of the $ \Lambda_1^{- 1} \Lambda_2^{- 1} \Lambda_1 \Lambda_2 $ commutant, where $ \Lambda_1 $, $ \Lambda_2 $ are random independent rotations with the distribution density $ \rho_0 $. It is shown that $ \rho_2 \equiv \sqrt{\rho_0 \rho_1} \frac{\pi \sqrt {2}} {4} $ and the measure $ \mu_1 $ with density $ \rho_1 $ is proportional to the basic measure $ \mu $.