Theorem. Equivariant elimination within each sector [fdip-0056]
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  • November 1, 2026

Let \(H=X\mathbin \perp Y\) be a finite-dimensional orthogonal real representation of a finite group \(G\), with both summands \(G\)-stable. Let the positive self-adjoint equivariant precision be \[Q=\begin {pmatrix}A&B\\B^*&C\end {pmatrix}, \qquad B:Y\to X.\] Then \(C>0\), and the retained precision \(S=A-BC^{-1}B^*:X\to X\) is positive, self-adjoint and \(G\)-equivariant. For every real irreducible type \(\Xi \), set \(X_\Xi =E_\Xi X\) and \(Y_\Xi =E_\Xi Y\). All block maps restrict to the indicated source and target sectors, and \[S|_{X_\Xi } =A|_{X_\Xi }-B_\Xi (C|_{Y_\Xi })^{-1}B_\Xi ^*.\] In common multiplicity coordinates, the same formula is \[S_\Xi =A_\Xi -B_\Xi C_\Xi ^{-1}B_\Xi ^*,\] where \(B_\Xi \) is an \(m_{X,\Xi }\times m_{Y,\Xi }\) matrix over \(D_\Xi \), and the products retain their displayed order.

Proof. 

  • November 1, 2026

Let \(\rho _X,\rho _Y\) denote the actions on the two summands. Equivariance of \(Q\) gives \(A\rho _X(g)=\rho _X(g)A\), \(B\rho _Y(g)=\rho _X(g)B\), and \(C\rho _Y(g)=\rho _Y(g)C\), together with the adjoint identity for \(B^*\). Positivity on \((0,y)\) makes \(C\) positive and invertible. Its inverse commutes with \(\rho _Y(g)\) by multiplying the commuting identity by \(C^{-1}\) on both sides. Hence the composite \(BC^{-1}B^*:X\to X\) and \(S\) commute with \(\rho _X(g)\). The same relations hold for the projectors, since they are real linear combinations of group actions.

Consequently \(B^*\) takes \(X_\Xi \) to \(Y_\Xi \), \(C^{-1}\) preserves \(Y_\Xi \), and \(B\) takes \(Y_\Xi \) back to \(X_\Xi \). The restriction of \(C^{-1}\) is the inverse of the restriction of \(C\), by restricting both inverse identities. This proves the typed formula, extending Proposition [fdip-003Q]. Choose isometric copies of the same real irreducible in \(X_\Xi \) and \(Y_\Xi \). Intertwiners then become rectangular matrices over \(D_\Xi \) and composition becomes ordered matrix multiplication, with adjunction given by conjugate transpose. This proves the multiplicity formula also over \(\mathbb H\); no scalar factors have been commuted. An absent \(Y_\Xi \) contributes a zero composite.

Completing the real square gives \[\begin {aligned} \langle (x,y),Q(x,y)\rangle &=\langle x,Sx\rangle \\ &\quad +\langle y+C^{-1}B^*x, C(y+C^{-1}B^*x)\rangle . \end {aligned}\] Testing at \(y=-C^{-1}B^*x\) proves \(S>0\); its self-adjointness follows by adjointing the composite. This uses the real inner product, regardless of the scalar algebra of a multiplicity matrix.

Use induced Euclidean volumes and let \(\pi :H\to X\) be the orthogonal projection. For every fixed \(x\), translation in \(Y\) and the completed square give \[\int _Y e^{-\langle (x,y),Q(x,y)\rangle/2}\,d\lambda _Y(y) =Z_Ce^{-\langle x,Sx\rangle/2}.\] Nonnegative Tonelli and normalization therefore give \(Z_Q=Z_CZ_S\) and \(\pi _*\mu _Q=\mu _S\), as in positive elimination. Applying the same argument in \(X_\Xi \mathbin \perp Y_\Xi \), and then the sector product law, gives \[\begin {aligned} (\mathcal D_X)_*\pi _*\mu _Q &=\bigotimes _\Xi \mu _{S|_{X_\Xi }},\\ (\pi _\Xi )_*\mu _{Q|_{H_\Xi }} &=\mu _{S|_{X_\Xi }},\qquad Z_C=\prod _\Xi Z_{C|_{Y_\Xi }}. \end {aligned}\] Here \(\pi _\Xi :H_\Xi \to X_\Xi \); the identity \(\pi E_\Xi =E_\Xi \pi \) shows that decomposing and projecting commute as maps. Product measures then give the same commuting square as in Theorem [fdip-003R], with all real isotypic sectors in place of the two sign sectors.

For vertex fields, the hypothesis holds when the retained set \(R\) is \(G\)-stable: take \(X=\mathbb R^R\) and \(Y=\mathbb R^{V\setminus R}\). It asserts equivariance of the induced marginal law. It does not assert that \(S=A\), the compressed precision. The cycle decimation gives a \(G\)-stable retained set with a nonzero Schur correction.