Theorem. Charged invariant tensors and multiplicity spaces [fdip-008B]
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  • November 1, 2026

Let \((R,\rho )\) be a specified finite-dimensional continuous unitary representation of the residual group \(K\) over \(\mathbb C\). Its complex linear dual \(R^*\) carries \((\rho ^*(g)\lambda )(v)=\lambda (\rho (g)^{-1}v)\). The charged physical space for this charge datum is \[\mathcal H_{\mathrm {phys}}(R) =(\mathcal H_{\mathrm {link}}\otimes R^*)^K,\] where \(g\) acts as \(R_g\otimes \rho ^*(g)\). Since \(R\) is finite dimensional, the Hilbert tensor product here is a finite orthogonal sum of copies of \(\mathcal H_{\mathrm {link}}\).

There is a canonical unitary identification \[\mathcal H_{\mathrm {phys}}(R) \cong \operatorname {Hom}_K(R,\mathcal H_{\mathrm {link}}) =:\mathcal M_R.\] Here \(\operatorname {Hom}_K\) consists of the linear maps \(F\) satisfying \(R_gF=F\rho (g)\) for every \(g\); its inner product is \(\langle F,G\rangle _{\mathrm {HS}}=\operatorname {Tr}(F^*G)\), linear in the second argument. The transfer operator \(T\otimes I\) preserves charged physical states and corresponds to \(F\mapsto TF\).

Suppose further that \(R\) is irreducible and put \(d=\dim R\). Its isotypic component in this complex Hilbert setting is \(\mathcal H_R\), defined as the closed span of all images \(F(R)\) with \(F\in \mathcal M_R\). Then \[\begin {aligned} R\otimes \mathcal M_R&\longrightarrow \mathcal H_R,\\ v\otimes F&\longmapsto \sqrt d\,F(v) \end {aligned}\] is a unitary representation isomorphism, with trivial \(K\)-action on \(\mathcal M_R\). The charged physical space is \(\mathcal M_R\), whereas \(\mathcal H_R\) includes the additional factor \(R\). In finite dimension, their dimensions are respectively \(\dim \mathcal M_R\) and \(d\dim \mathcal M_R\).

Proof. 

  • November 1, 2026

For an elementary tensor define \[\Phi (x\otimes \lambda )(v)=\lambda (v)x.\] If \(e_1,\ldots ,e_d\) is an orthonormal basis of \(R\) with dual basis \(e_1^*,\ldots ,e_d^*\), the inverse is \[F\longmapsto \sum _{i=1}^dF(e_i)\otimes e_i^*.\] Evaluating on the basis proves both inverse identities. The map \(\Phi \) is basis independent, so its inverse is as well. Its norm is the Hilbert–Schmidt norm \(\|F\|_{\mathrm {HS}}^2=\sum _i\|F(e_i)\|^2\), proving that \(\Phi \) is unitary.

Under \(\Phi \), the tensor action becomes \[F\longmapsto R_gF\rho (g)^{-1}.\] Indeed, apply both sides to \(v\) on an elementary tensor. Its fixed points are exactly the intertwiners, which proves the charged identification. The invariant subspace is closed, so \(\mathcal M_R\) is a Hilbert space. Since \(TR_g=R_gT\), postcomposition by \(T\) preserves intertwiners and corresponds to \(T\otimes I\).

For irreducible \(R\), unitarity and the intertwining identities show that \(F^*G:R\to R\) commutes with \(\rho (K)\). Complex Schur's lemma gives \[F^*G=\frac {\operatorname {Tr}(F^*G)}d I_R =\frac {\langle F,G\rangle _{\mathrm {HS}}}d I_R.\] One elementary proof of the lemma chooses an eigenvalue of a commuting complex endomorphism: its nonzero invariant eigenspace must be all of an irreducible representation. Therefore \[\langle F(v),G(w)\rangle =\frac 1d\langle v,w\rangle \langle F,G\rangle _{\mathrm {HS}}.\] Extending by linearity in finite sums proves that \(\sqrt d\) times evaluation is isometric. Its range is closed, and contains every \(F(R)\); by the definition of \(\mathcal H_R\), it is onto. The intertwining identity makes it equivariant. This proves the stated isotypic factorization and its normalization.

The algebraic tensor-to-map equivalence is formalized by Representation.Equiv.dualTensorHom: its source order is \(R^*\otimes \mathcal H_{\mathrm {link}}\), so first swap our two factors. It requires finite-dimensional \(R\), with no dimension restriction on \(\mathcal H_{\mathrm {link}}\). The fixed-map step is formalized by Representation.invariantsEquivIntertwiningMap for the conjugation action on linear maps. The Hilbert–Schmidt isometry and \(\sqrt d\) normalization are the direct arguments above. The earlier real isotypic projectors and multiplicity matrices concern whole representation components; a projector onto such a component does not by itself impose the charged tensor constraint.

For reducible charge data, first choose an irreducible orthogonal decomposition \(R=\bigoplus _\lambda R_\lambda \otimes \mathbb C^{n_\lambda }\). Applying the intertwiner identification to each summand gives \[\mathcal M_R\cong \bigoplus _\lambda \operatorname {Hom}(\mathbb C^{n_\lambda },\mathcal M_{R_\lambda }).\] This is an isomorphism of vector spaces. If all these spaces are finite dimensional and \(m_\lambda =\dim \mathcal M_{R_\lambda }\), then \(\dim \mathcal M_R=\sum _\lambda n_\lambda m_\lambda \). For the trivial one-dimensional charge, the construction recovers the neutral space.