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FROBENIUS–SCHUR INDICATORS FOR PROJECTIVE CHARACTERS WITH APPLICATIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 111 / Issue 2 / April 2025
- Published online by Cambridge University Press:
- 08 October 2024, pp. 290-297
- Print publication:
- April 2025
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- Article
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Let
$\alpha $ be a complex valued 
$2$-cocycle of finite order of a finite group 
$G.$ The nth Frobenius–Schur indicator of an irreducible 
$\alpha $-character of G is defined and its properties are investigated. The indicator is interpreted in general for 
$n =2$ and it is shown that it can be used to determine whether an irreducible 
$\alpha $-character is real-valued under the assumption that the order of 
$\alpha $ and its cohomology class are both 
$2$. A formula, involving the real 
$\alpha $-regular conjugacy classes of 
$G,$ is found to count the number of real-valued irreducible 
$\alpha $-characters of G under the additional assumption that these characters are class functions.
