Theorem. the Frobenius--Schur trichotomy [fgap-0021]
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frobeniusSchurIndicator_eq_one_or_eq_zero_or_eq_neg_onefrobeniusSchurIndicator_eq_one_ifffrobeniusSchurIndicator_eq_neg_one_ifffrobeniusSchurIndicator_eq_zero_iff
L∃∀N

    Let \(\rho \) be an irreducible finite-dimensional representation of a finite group over an algebraically closed field of characteristic 0. Its Frobenius--Schur indicator \[ \nu _2(\rho )=\frac 1{|G|}\sum _{g\in G}\chi _\rho (g^2) \] takes exactly 1 of the values \(1\), \(0\), and \(-1\). The value \(1\) is equivalent to the existence of a nondegenerate invariant symmetric bilinear form; \(-1\) is equivalent to a nondegenerate invariant alternating form; and \(0\) is equivalent to the absence of a nonzero invariant bilinear form. See [james2001representations, def. 23.13, the proof of thm. 23.14, and thm. 23.16, pp. 273--277]. James and Liebeck state the alternatives using nonzero invariant forms; for an irreducible representation, a nonzero invariant symmetric or alternating form has zero radical and is therefore nondegenerate.

    This trichotomy classifies the invariant-form behavior of the complex irreducible representation. It does not by itself compute a Schur index or a complete decomposition after descent to \(\mathbb R\).