### Involutions in finite groups and the Frobenius-Schur indicator of characters

Algebra and Geometry Seminar

30th January 2019, 2:30 pm – 3:30 pm

Howard House, 4th Floor Seminar Room

The Frobenius-Schur indicator of a real-valued irreducible complex group character equals 1 or -1, but which value occurs seems at times almost to be a random process. In all examples we know, there are more characters with indicator value equal to 1 than with value -1, but nothing definitive is proved yet. The following concepts seem to be relevant. A group element is said to be real if it is conjugate to its inverse. The element is strongly real if it is inverted by an involution. We say that a real element is weakly real if it is not strongly real. The main theme of the talk is to indicate the influence of weakly real elements of odd order in constructing real-valued irreducible characters with indicator -1.

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