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?
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.