Corollary. Which symmetry selectors restrict [fdip-005N]
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  • November 1, 2026

Let a finite group \(\Gamma \) act orthogonally on a finite-dimensional real Euclidean space \(H\), and let \(Q>0\) be self-adjoint and \(\Gamma \)-equivariant. Two constructions of symmetry selectors have different criteria.

  1. For a self-adjoint idempotent \(e\in \mathbb R[\Gamma ]\), the selector \(P=\rho (e)\) restricts whenever its rank is nonzero and proper. Its image need not be a \(\Gamma \)-subrepresentation.
  2. A nonzero proper \(\Gamma \)-subrepresentation \(X\) restricts precisely when its multiplicity subspace in every real isotypic block is invariant under the corresponding Hermitian precision matrix \(Q_\Xi \).

Here is the precise multiplicity convention. Use the real commutant and the right-scalar column coordinates. For \(D_\Xi =\operatorname {End}_\Gamma (W_\Xi )\) and \(r_\Xi =\dim _{D_\Xi }W_\Xi \), represent \(H_\Xi \) by \(r_\Xi \) columns in \(D_\Xi ^{m_\Xi }\); matrices act on each column by left multiplication. There is a right \(D_\Xi \)-subspace \(U_\Xi \subseteq D_\Xi ^{m_\Xi }\) such that \[X\cap H_\Xi \cong \{(z_1,\ldots ,z_{r_\Xi }):z_a\in U_\Xi \}.\] The criterion is \[Q_\Xi U_\Xi \subseteq U_\Xi \qquad \text {for every }\Xi .\] Equivalently, each \(U_\Xi \) has a \(D_\Xi \)-basis of \(Q_\Xi \)-eigenvectors. Repeated eigenvalues allow proper subspaces of an eigenspace; whole eigenspaces need not be selected.

Proof. 

  • November 1, 2026

Equivariance makes \(Q\) commute with every \(\rho (g)\), hence with \(\rho (e)\). The restriction criterion proves the first assertion. Centrality is not needed for this commutation; it would additionally make \(\rho (e)\) commute with the group action.

For a subrepresentation \(X\), both \(X\) and \(X^\perp \) are \(\Gamma \)-stable, so its orthogonal projector belongs to the commutant. By Theorem [fdip-004Z] it is represented by Hermitian idempotents \(p_\Xi \in M_{m_\Xi }(D_\Xi )\). In the stated column coordinates its image has exactly the displayed form with \(U_\Xi =\operatorname {im}p_\Xi \). Conversely a right \(D_\Xi \)-subspace has an orthogonal projector commuting with right scalar multiplication: its real orthogonal complement is right-scalar stable, since multiplication by each unit scalar is orthogonal and preserves the subspace. Thus that projector is a matrix over \(D_\Xi \); the commutant identification gives the corresponding subrepresentation.

The matrix \(Q_\Xi \) acts on every column, so \(QX\subseteq X\) is exactly the displayed invariance condition. If it holds, self-adjointness preserves the orthogonal complement too. The Hermitian spectral argument of Corollary [fdip-0050] applies to the restriction on \(U_\Xi \): a real eigenvector generates a right \(D_\Xi \)-line of eigenvectors, its orthogonal complement is invariant, and induction gives a \(D_\Xi \)-orthonormal eigenbasis. Conversely a span of eigenvectors is invariant, since their eigenvalues are real and the matrix is right-linear. Apply Theorem [fdip-005M] to finish.

In a multiplicity-free representation every \(m_\Xi =1\), and a right \(D_\Xi \)-subspace of \(D_\Xi \) is either zero or all of it. Thus every nonzero proper subrepresentation restricts. At higher multiplicity, group stability alone does not suffice, as the prism line shows.