Higher Frobenius-Schur Indicators
Just as for any complex representation ρ,
is a self-intertwiner, for any integer n,
is also a self-intertwiner. By Schur's lemma, this will be a multiple of the identity for irreducible representations. The trace of this self-intertwiner is called the nth Frobenius-Schur indicator.
The original case of the Frobenius-Schur indicator is that for n = 2. The zeroth indicator is the dimension of the irreducible representation, the first indicator would be 1 for the trivial representation and zero for the other irreducible representations.
It resembles the Casimir invariants for Lie algebra irreducible representations. In fact, since any rep of G can be thought of as a module for C and vice versa, we can look at the center of C. This is analogous to looking at the center of the universal enveloping algebra of a Lie algebra. It is simple to check that
belongs to the center of C, which is simply the subspace of class functions on G.
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