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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?

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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:

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

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  • $\begingroup$ Thank you very much for the papers! They look quite interesting and put things much more in the context of my background. Thanks also to the link to your question--I had not found those resources yet. Is there any specifics on the irreps of $SO(3)$ or am I just chasing an imaginary cat? $\endgroup$ – Mortified Through Math Jan 4 '16 at 6:44

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