Editing Frobenius–Schur indicator (section)

==Higher Frobenius-Schur indicators==
Just as for any complex representation ρ,

:<math>\frac{1}{|G|}\sum_{g\in G}\rho(g)</math>

is a self-intertwiner, for any integer ''n'',

:<math>\frac{1}{|G|}\sum_{g\in G}\rho(g^n)</math>

is also a [[intertwiner|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 n<sup>th</sup> ''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 invariant]]s for [[Lie algebra]] irreducible representations. In fact, since any representation of G can be thought of as a [[module (mathematics)|module]] for '''C'''[''G''] and vice versa, we can look at the [[center (algebra)|center]] of '''C'''[''G'']. This is analogous to looking at the center of the [[universal enveloping algebra]] of a Lie algebra. It is simple to check that

:<math>\sum_{g\in G}g^n</math>

belongs to the center of '''C'''[''G''], which is simply the subspace of class functions on ''G''.