Proposition. From Frobenius-Schur indicators to real Gaussian types [fdip-0051]
✍️source

  • November 1, 2026

Let \(W\) be an irreducible real representation of a finite group, \(D=\operatorname {End}_G(W)\), and \(\chi \) an irreducible constituent of \(W_{\mathbb C}=W\otimes _{\mathbb R}\mathbb C\). The Frobenius-Schur indicator gives exactly the following possibilities, where \(X_\chi \) denotes a complex module of character \(\chi \): \[\begin {array}{c|c|c|c} \nu _2(\chi )&D&W_{\mathbb C}&\dim _{\mathbb R}W\\\hline 1&\mathbb R&X_\chi &\chi (1)\\ 0&\mathbb C&X_\chi \oplus X_{\bar \chi }&2\chi (1)\\ -1&\mathbb H&X_\chi \oplus X_\chi &2\chi (1). \end {array}\] In the middle row \(\chi \ne \bar \chi \). In the last row \(\chi (1)\) is even and \(\dim _{\mathbb H}W=\chi (1)/2\). The form trichotomy alone is not the real descent argument; these assertions use the bridge proved below. They agree with [serre1977linear, Section 13.2, Proposition 39, p. 109].

Proof. 

  • November 1, 2026

Commutation with the finitely many real group matrices is a system of real linear equations. Complexifying its solution space therefore gives \[\operatorname {End}_G(W_{\mathbb C}) \cong D\otimes _{\mathbb R}\mathbb C.\] For \(D=\mathbb R,\mathbb C,\mathbb H\) the right side is respectively \(\mathbb C\), \(\mathbb C\times \mathbb C\), \(M_2(\mathbb C)\). For the middle identity use \(z\otimes a\mapsto (za,\bar z a)\); for the last use the usual quaternion two-by-two complex matrices, whose complex span is all \(M_2(\mathbb C)\), as in Proposition [fcap-0042]. Complex semisimplicity identifies the left side with \(\prod _\psi M_{n_\psi }(\mathbb C)\), where \(n_\psi \) is the constituent multiplicity. Its simple factors and their dimensions force one constituent once, two distinct constituents once each, or one constituent twice, respectively.

Let \(c(w+iv)=w-iv\). It exchanges the \(\psi \) and \(\bar \psi \) components. If \(D=\mathbb R\), the single complex irreducible has the real form \(W\). Averaging a real inner product and complexifying it supplies an invariant nondegenerate symmetric bilinear form, so \(\nu _2=1\). If \(D=\mathbb C\) and either of the two inequivalent components were fixed by \(c\), its fixed vectors would be a nonzero proper real invariant subspace of \(W\). Indeed, for a \(c\)-stable space every vector is uniquely \((x+cx)/2+i(x-cx)/(2i)\) with both components fixed by \(c\). Irreducibility excludes this, so the two characters are a nonreal conjugate pair and \(\nu _2=0\).

If \(D=\mathbb H\), conjugation fixes the sole constituent type, so \(\chi =\bar \chi \) and \(\nu _2=\pm 1\). Suppose \(\nu _2=1\). Choose a nondegenerate invariant symmetric complex bilinear form \(B\) on \(X_\chi \) and an averaged invariant positive Hermitian form \(h\), conjugate-linear in its first argument. There is a unique antilinear map \(J\) with \(B(x,y)=h(Jx,y)\), and it commutes with \(G\). In an \(h\)-orthonormal basis write \(B(x,y)=x^TAy\). Then \(A^T=A\), \(Jx=\bar A\bar x\) and \(J^2=\bar A A=A^*A>0\). By complex Schur's lemma \(J^2=tI\) for a positive real \(t\). Replacing \(J\) by \(t^{-1/2}J\) makes \(J^2=I\). Its fixed space \(U\) is a real form of \(X_\chi \), by the same two-component formula. This is the irreducible specialization of [serre1977linear, Section 13.2, Theorem 31, pp. 106-108].

The real module \(U\) is irreducible, since a proper real invariant subspace would complexify to a proper subspace of \(X_\chi \). There is a nonzero complex intertwiner \(U_{\mathbb C}\to W_{\mathbb C}=X_\chi ^{\oplus 2}\). In real bases its real or imaginary part gives a nonzero real intertwiner \(U\to W\), which must be an isomorphism by irreducibility. But \(\dim _{\mathbb R}U=\chi (1)\) and \(\dim _{\mathbb R}W=2\chi (1)\), a contradiction. Thus \(\nu _2=-1\). The three division algebras exhaust the cases and give distinct indicators, so all implications reverse. Finally \(W\) is a left \(\mathbb H\)-space, making its real dimension a multiple of 4; hence \(\chi (1)\) is even.

For an equivariant centered real Gaussian on an isotypic sector, use the coordinates \(z_1,\ldots ,z_r\in D^m\) of Corollary [fdip-0050]. Its real density is proportional to \[\exp \left (-\frac 12\sum _{a=1}^r \operatorname {Re}(z_a^*Q_\Xi z_a)\right ).\] The real covariance is the Hermitian multiplicity matrix \(Q_\Xi ^{-1}\) acting on each column. For \(D=\mathbb C\), multiplication of a column by \(e^{i\theta }\) preserves this real law: this is circular symmetry. With the displayed real normalization the complex second moment is \(\mathbb E[z_a z_a^*]=2Q_\Xi ^{-1}\). For \(D=\mathbb H\), right multiplication \(z_a\mapsto z_a q\) by a unit quaternion likewise preserves the real quadratic form and Lebesgue measure. This is the asserted quaternionic symmetry; left multiplication by an arbitrary quaternion need not commute with the entries of \(Q_\Xi \). These coordinate symmetries need not be additional graph symmetries. The real Gaussian normalization and covariance are those of Theorem [fdip-000H].

TauCeti's Representation.frobeniusSchurIndicator_eq_one_iff_isRealizableOverReal proves the indicator-one/real-realizability equivalence for an irreducible complex representation of a finite group. It supports the real-form step; the full commutant identification, all three real dimensions and the Gaussian coordinate statements are the arguments above.