Q:

Applications of $SO(3)$ irreps to spatial rotation

I've been on a kick learning about Lie Groups, with special emphasis on $SO(3)$ recently. I work in the field of spacecraft attitude determination and control, where is $SO(3)$ of interest in the literal sense of capturing the rotational orientation of a spacecraft, and have been studying Lie theory to try and get a better handle on the bigger "whys" of the actual parameterizations we use. I notice there's a lot of emphasis on irreps of Lie Groups, especially in the physics literature, but all of the examples I run across apply irreps to calculating allowed eigenstates in quantum systems. Are there any applications of irreps to more direct and mundane problems like representing or determining spatial orientation? 

A:

The simple answer is that high-performance attitude filters can be constructed which exploit the the properties of lie groups. For example, see the following papers and citations thereof:

Mahony, R.; Hamel, T. & Pflimlin, J.-M. Nonlinear Complementary Filters on the Special Orthogonal Group, IEEE Transactions on Automatic Control, 2008, 53, 1203-1218
Hertzberg, C.; Wagner, R.; Frese, U. & Schröder, L. Integrating generic sensor fusion algorithms with sound state representations through encapsulation of manifolds, Information Fusion, 2013, 14, 57 - 77

The following question might also be on interest to you if you are just starting out with lie groups.

