I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator.
$\bullet$ What is the important use of Frobenius–Schur indicators? Why such a notion is important? What are examples that using Frobenius–Schur indicators can help us classifying some vector spaces or (discrete) groups?
$\bullet$ How about Frobenius-Schur exponents(e.g. here and here)?
It seems to me that if I understand that the concept of discrete Fourier transformation for the finite group, the Frobenius–Schur indicators to be some-sort of invariance or measure.
$\bullet$ It is said that Frobenius-Schur indicators have values of 0, +1, -1; but I find this paper define the higher Frobenius-Schur indicators $\nu^{(n)}$, which has values larger than $1$, such as $\nu^{(n)}=2$ in the page 22's Appendix. Can someone address why this is the case that the higher Frobenius-Schur indicators $\nu^{(n)}$ may not be 0, +1, -1?
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p.s. I had learned that Frobenius-Schur indicators/exponents are useful to classify certain Spherical Categories. Maybe someone can also say a few words about this direction?