Theorem. The real commutant and multiplicity matrices [fdip-004Z]
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  • November 1, 2026

Let a finite group \(G\) act orthogonally on a finite-dimensional real space \(H\). In its isotypic decomposition, write \(H_\Xi \cong W_\Xi ^{\oplus m_\Xi }\), where \(W_\Xi \) is irreducible and \(m_\Xi >0\) is its multiplicity. The commutant consists of all linear maps commuting with \(G\). With multiplication given by composition, put \(D_\Xi =\operatorname {End}_G(W_\Xi )\). Then \[\begin {aligned} D_\Xi &\cong \mathbb R,\ \mathbb C\ \text {or}\ \mathbb H,\\ \operatorname {End}_G(H)&\cong \prod _\Xi M_{m_\Xi }(D_\Xi ). \end {aligned}\] The isomorphism can be chosen to carry adjunction to conjugate transpose. If \(H=\mathbb R^V\) is a vertex permutation representation, the real dimension of \(\operatorname {End}_G(H)\) is the number of orbits of the diagonal action \(g(u,v)=(gu,gv)\) on \(V\times V\). These orbits are called orbitals.

Proof. 

  • November 1, 2026

Maschke's theorem decomposes \(H\) orthogonally into irreducibles. The kernel and image of an intertwiner are invariant. Hence an intertwiner between inequivalent irreducibles is zero, and every nonzero endomorphism of one irreducible is invertible. Thus \(D_\Xi \) is a finite-dimensional associative real division algebra; Frobenius's division-algebra theorem gives the three displayed possibilities. Between \(m_\Xi \) identified copies the intertwiners are precisely matrices of elements of \(D_\Xi \), acting by \((Aw)_i=\sum _j A_{ij}(w_j)\). This proves the algebra statement. The real classification and its representation-theoretic use are [serre1977linear, Section 13.2, pp. 106-109]; compare the real-type distinction.

Adjunction preserves \(D_\Xi \). A self-adjoint element of it has real eigenspaces invariant under \(G\), so irreducibility makes it a real scalar. In particular \(d^*d=cI\) with \(c>0\) for \(d\ne 0\). The resulting positive adjoint on \(\mathbb R\), \(\mathbb C\) or \(\mathbb H\) is the usual conjugation: for an imaginary unit \(j^2=-1\), write \(j+j^*=2aI\) with \(a\in \mathbb R\). Then \(j^*j=2aj+I\) is scalar, forcing \(a=0\). Thus \(j^*=-j\). Identifications between equivalent orthogonal copies can be made isometric by dividing an intertwiner by the positive square root of its scalar \(T^*T\). In those coordinates the block adjoint is conjugate transpose.

For vertex permutations, a matrix \(A\) commutes with every \(\rho (g)\) exactly when \(A_{gu,gv}=A_{u,v}\). Its entries are therefore constant on orbitals. The indicator matrices of distinct orbitals have disjoint supports and form a real basis of the commutant, proving the dimension formula.

The scalar-side convention matters for \(\mathbb H\). Here \(D_\Xi \) acts on \(W_\Xi \) on the left by endomorphisms. If instead \(W_\Xi \) is presented as a left module for a simple matrix algebra, its commuting scalars act on the right and form an opposite algebra, as in Proposition [fcap-0042]. Quaternion conjugation identifies that opposite algebra with \(\mathbb H\); it does not license reversing the order of quaternion products.