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+\section{Symmetry-adapted block decomposition of the shadow map}
+\label{sec:symmetry-blocks}
+ 
+The constructions of Sections~\ref{sec:response-maps}--\ref{sec:multiparty-sources} treat the sector grading $V\subseteq S$, $T\subseteq S^c$ as the only available organizing structure on either side of a cut. When $\rho$ carries an additional symmetry --- exactly, or after an appropriate twirl --- this grading can be refined further, along representation-theoretic rather than combinatorial lines. The refinement is worth having for four distinct reasons, which we state before developing the formal statements, since they motivate different parts of what follows and are not equally strong.
+ 
+\begin{itemize}
+\item \emph{Explanatory power.} Several numerical facts already reported in this note --- the triple degeneracy of the singular values of $\mathcal M_a$ for $\GHZ_3$ (Section~\ref{sec:qubit-numerics}), the triple degeneracy at the Smolin state under both the $1\mid3$ and $2\mid2$ cuts (Section~\ref{sec:multiparty-sources}), and the rigid common value across all $38$ four-qubit graph states --- have so far been recorded as numerical observations. Proposition~\ref{prop:block-diagonal} below shows that such degeneracies are not coincidental: whenever the source side of a cut is symmetry-irreducible (offers only one isotypic component under the relevant group), the corresponding degeneracy is forced. This converts a family of separately verified numerical facts into one structural statement.
+\item \emph{Diagnostic resolution.} The sub-block witnesses of Corollary~\ref{cor:sub-block} resolve a violation only down to the level of \emph{which parties} are involved ($V,T$). The isotypic decomposition resolves it further, down to \emph{which symmetry channel} within a fixed $(V,T)$ sector is responsible --- for instance, whether a two-party source correlation block is carrying its signal in a totally symmetric or in an antisymmetric combination of its constituents. This is invisible to the party-indexed grading alone.
+\item \emph{Computational cost.} For a source or target cluster respecting a symmetry group $G$, the isotypic decomposition replaces one singular value decomposition on the full sector space by several independent, much smaller singular value decompositions on the multiplicity spaces $M_\lambda$, whose dimensions grow far more slowly than the ambient sector dimension as the cluster size increases. This is the natural computationally tractable foothold for the symmetric sub-family of the higher-order tensor construction discussed in the closing remark of Section~\ref{sec:tensor-viewpoint}.
+\item \emph{A one-sided extension via twirling.} If $\rho$ itself lacks the symmetry but a twirl $T_G(\rho)$ is cheap to evaluate, entanglement detected on $T_G(\rho)$ certifies entanglement of $\rho$ (twirling by local unitaries preserves separability), so the block-diagonal criterion below can serve as an inexpensive pre-test, with the standard one-sided caveat that a negative result on $T_G(\rho)$ is uninformative about $\rho$.
+\end{itemize}
+ 
+What this refinement does \emph{not} deliver is a numerically sharper detection threshold: Remark~\ref{rem:no-universal-sharpening} below shows that no universal, state-independent improvement over Theorems~\ref{thm:cut-bound} and \ref{thm:cluster-cut} exists at the level of an individual block, except in the single-isotype case where the improvement is one of concentration rather than of threshold value.
+ 
+\subsection*{Setup}
+ 
+Let $G$ be a compact group acting on the system by local unitaries, $g\mapsto \bigotimes_{a\in P}U_g^{(a)}$, and suppose $\rho$ is $G$-invariant: $(\bigotimes_a U_g^{(a)})\,\rho\,(\bigotimes_a U_g^{(a)})^\dagger=\rho$ for all $g\in G$. Fix a cut $S\mid S^c$ preserved by $G$ as a set partition, so that $G$ acts on $\V_0^{(S)}$ and on $\V_0^{(S^c)}$ separately, via the adjoint representations $\mathrm{Ad}^{(S)}_g$, $\mathrm{Ad}^{(S^c)}_g$. Decompose both traceless spaces into isotypic components,
+\begin{equation}
+\V_0^{(S)}=\bigoplus_\lambda M_\lambda^{(S)}\otimes V_\lambda,
+\qquad
+\V_0^{(S^c)}=\bigoplus_\lambda M_\lambda^{(S^c)}\otimes V_\lambda,
+\label{eq:isotypic-decomp}
+\end{equation}
+where $\lambda$ ranges over the irreducible representations of $G$ appearing on either side, $V_\lambda$ denotes a fixed model of the irreducible representation of dimension $d_\lambda$, and $M_\lambda^{(S)}$, $M_\lambda^{(S^c)}$ are the corresponding multiplicity spaces. This is exactly the same type of decomposition already used for the source and target sector gradings in Sections~\ref{sec:response-maps}--\ref{sec:multiparty-sources}, now taken with respect to a representation-theoretic rather than a combinatorial grading; the two coincide only in the abelian, source-cluster-trivial special case worked out numerically at the start of this section.
+ 
+\begin{proposition}[Exact block-diagonality]
+\label{prop:block-diagonal}
+Under the hypotheses above, $\widetilde{\mathcal M}_S(\rho)$ is block diagonal with respect to the isotypic decomposition~\eqref{eq:isotypic-decomp}: writing $\Pi_\lambda^{(S^c)}$, $\Pi_\mu^{(S)}$ for the isotypic projections,
+\begin{equation*}
+\Pi_\lambda^{(S^c)}\,\widetilde{\mathcal M}_S(\rho)\,\Pi_\mu^{(S)}=0
+\qquad\text{whenever }\lambda\neq\mu,
+\end{equation*}
+exactly, not merely approximately or in a bound. Moreover, on the surviving diagonal blocks,
+\begin{equation}
+\widetilde{\mathcal M}_S(\rho)\big|_\lambda = A_\lambda\otimes\mathrm{id}_{V_\lambda}
+\label{eq:wigner-eckart}
+\end{equation}
+for a unique linear map $A_\lambda:M_\lambda^{(S)}\to M_\lambda^{(S^c)}$, the reduced shadow map at $\lambda$.
+\end{proposition}
+ 
+\begin{proof}
+$G$-invariance of $\rho$ gives $\mathrm{Ad}^{(S^c)}_g\,\widetilde{\mathcal M}_S(\rho)=\widetilde{\mathcal M}_S(\rho)\,\mathrm{Ad}^{(S)}_g$ for all $g$, i.e.\ $\widetilde{\mathcal M}_S(\rho)$ is a $G$-equivariant map between the two representations~\eqref{eq:isotypic-decomp}. Both claims are then Schur's lemma applied to the isotypic decomposition: equivariant maps vanish between inequivalent irreducible summands, and act as a fixed scalar multiple of the identity on the irreducible factor $V_\lambda$ within a matching pair, leaving exactly the freedom recorded in $A_\lambda$ on the multiplicity spaces.
+\end{proof}
+ 
+\subsection*{Exact formula and its consequence}
+ 
+\begin{proposition}[Reduced matrix element formula]
+\label{prop:reduced-formula}
+Let $\rho=T_G(\rho_S\otimes\sigma_{S^c})$ be the $G$-twirl of a product state across $S\mid S^c$, so $\rho$ is separable (a mixture, over $g\in G$, of product states) and $G$-invariant. Let $r^{(S)}\in\V_0^{(S)}$, $v_{S^c}\in\V_0^{(S^c)}$ be the traceless Bloch vectors of $\rho_S$, $\sigma_{S^c}$, with isotypic components $r_\lambda$, $v_\lambda$. Then
+\begin{equation}
+A_\lambda=\frac{1}{d_\lambda}\,\tilde v_\lambda\,\tilde r_\lambda^{\,T},
+\label{eq:reduced-element}
+\end{equation}
+where $\tilde r_\lambda$, $\tilde v_\lambda$ are $r_\lambda$, $v_\lambda$ reshaped as $(\dim M_\lambda^{(S)})\times d_\lambda$ and $(\dim M_\lambda^{(S^c)})\times d_\lambda$ matrices in a basis of $V_\lambda$ shared by both sides. Consequently,
+\begin{equation}
+\dim(V_\lambda)\,\norm{A_\lambda}_*\;\le\;\norm{r_\lambda}\,\norm{v_\lambda}.
+\label{eq:per-block-bound}
+\end{equation}
+\end{proposition}
+ 
+\begin{proof}
+For fixed $g$, the product term $(\mathrm{Ad}^{(S)}_g r^{(S)})(\mathrm{Ad}^{(S^c)}_g v_{S^c})^T$ (unnormalized) is rank one, exactly as in the proof of Theorem~\ref{thm:cluster-cut}. Averaging over $g$ and expanding both factors in the isotypic bases gives, by the Schur orthogonality relation $\int_G D^\lambda(g)_{cb}D^\lambda(g)_{da}\,dg=\tfrac1{d_\lambda}\delta_{cd}\delta_{ab}$ for the (real, orthogonal) irreducible matrix elements $D^\lambda$, exactly Eq.~\eqref{eq:reduced-element} on each isotypic block, with all cross-$\lambda$ contributions vanishing by the same orthogonality relation applied to inequivalent irreducibles. Equation~\eqref{eq:per-block-bound} then follows from the standard nuclear-norm bound $\norm{AB}_*\le\norm{A}_\fro\norm{B}_\fro$ applied to Eq.~\eqref{eq:reduced-element}, using $\norm{\tilde r_\lambda}_\fro=\norm{r_\lambda}$, $\norm{\tilde v_\lambda}_\fro=\norm{v_\lambda}$.
+\end{proof}
+ 
+\begin{corollary}[Consistency with the cut-separable bound]
+\label{cor:consistency}
+Under the hypotheses of Proposition~\ref{prop:reduced-formula},
+\begin{equation}
+\sum_\lambda \dim(V_\lambda)\,\norm{A_\lambda}_*
+\;\le\;
+\norm{r^{(S)}}\,\norm{v_{S^c}}
+\;\le\;
+\sqrt{(d_S-1)(d_{S^c}-1)},
+\label{eq:cs-recovery}
+\end{equation}
+the first inequality by Cauchy--Schwarz over $\lambda$ applied to Eq.~\eqref{eq:per-block-bound}, and the second by the correlation-sum identity used in Theorem~\ref{thm:cluster-cut}. By linearity and convexity of the nuclear norm, Eq.~\eqref{eq:cs-recovery} extends to arbitrary $G$-invariant separable states (finite or continuous mixtures of twirled product terms), recovering the bound of Theorem~\ref{thm:cluster-cut} through the block decomposition rather than around it.
+\end{corollary}
+ 
+\begin{remark}[No universal per-block sharpening, except by concentration]
+\label{rem:no-universal-sharpening}
+The first inequality in Eq.~\eqref{eq:cs-recovery} is generically strict: equality in Cauchy--Schwarz requires $\norm{r_\lambda}\propto\norm{v_\lambda}$ across all $\lambda$, which independently chosen $\rho_S,\sigma_{S^c}$ have no reason to satisfy. A direct numerical check on a twirled random product state gives $\norm{\mathcal M_S(\rho)}_*=0.429$ against the bound $\norm{r^{(S)}}\norm{v_{S^c}}=0.750$ from Eq.~\eqref{eq:cs-recovery} --- a strict, and generic, gap. Since Eq.~\eqref{eq:per-block-bound} bounds $\norm{r_\lambda}$ only by the global $\norm{r^{(S)}}^2\le d_S-1$, with no constraint on how the purity budget distributes across $\lambda$ for a generic $\rho_S$, no universal constant improving on $\sqrt{(d_S-1)(d_{S^c}-1)}$ holds for an individual block $\lambda$ in general.
+ 
+The exception is exactly the case relevant to Sections~\ref{sec:response-maps}--\ref{sec:multiparty-sources}: if $\V_0^{(S)}$ itself carries only a single isotypic component under $G$ --- as happens automatically for a singleton qubit source, whose traceless Bloch space is irreducible under collective $SU(2)$ --- then $r_\lambda=r^{(S)}$ trivially, the full purity budget sits in the one available block, and Eq.~\eqref{eq:per-block-bound} becomes
+\begin{equation*}
+\dim(V_\lambda)\,\norm{A_\lambda}_*\;\le\;\sqrt{(d_S-1)(d_{S^c}-1)},
+\end{equation*}
+numerically identical to Theorem~\ref{thm:cut-bound}, but now a statement about a matrix of size $\dim M_\lambda^{(S^c)}\times1$ rather than the full target space. This is precisely the mechanism forcing the triple degeneracies recorded for $\GHZ_3$ and the Smolin state: a single-isotype source leaves the reduced map $A_\lambda$ no room to be anything but a vector, so $\widetilde{\mathcal M}_S(\rho)$ can only ever have $\dim(V_\lambda)$ equal singular values, regardless of which particular symmetric state is chosen.
+\end{remark}
+ 
+\begin{remark}[Structural gradings beyond the chosen group]
+A numerical check on a state built from a real (in the computational basis) combination of permutation operators shows a further, unforced splitting inside a single $SU(2)$-isotypic block: sub-blocks distinguished by the parity of the number of $Y$-generators in the underlying Pauli string decouple exactly, because $\sigma_y^T=-\sigma_y$ forces $\tr(\rho\bigotimes_a\sigma_{i_a})=0$ whenever $\rho$ is real and the string contains an odd number of $y$-indices. This $\mathbb Z_2$ grading commutes with every construction in this note, costs nothing to check, and refines the block structure further whenever $\rho$ happens to be real in the chosen basis --- which holds for every graph state, $\GHZ_n$, $W_n$, Dicke state, and the Smolin state used elsewhere in this note.
+\end{remark}