M.7 Covariance and Ward-compatibility constraints

with the variational subspace. Let $\mathrm{Equiv}\subset {\mathcal{C}}_{\mathrm{coh}}$ be the $14$-dimensional span of the corrected $C_{k,j}$ and $V:=\mathrm{Equiv}\cap \mathrm{SA}$ (reading-dependent). Write ${\mathcal{C}}_{\mathrm{coh}}={\mathcal{C}}_{\mathrm{K}}\oplus {\mathcal{C}}_{\mathrm{reg}}$ by grade ($N\le 2$ / $N\ge 3$), with projections ${\pi }_K,{\pi }_{\mathrm{reg}}$. Four distinct objects are reported: the projections ${\pi }_K(V)$, ${\pi }_{\mathrm{reg}}(V)$ (equivariant, but not self-adjoint in general) and the intersections $V\cap {\mathcal{C}}_{\mathrm{K}}$, $V\cap {\mathcal{C}}_{\mathrm{reg}}$ (genuine SA assignments). Always $\dim (V\cap {\mathcal{C}}_{\mathrm{K}})=\dim V-\dim {\pi }_{\mathrm{reg}}(V)$, $\dim (V\cap {\mathcal{C}}_{\mathrm{reg}})=\dim V-\dim {\pi }_K(V)$, hence $\dim (V\cap {\mathcal{C}}_{\mathrm{K}})+\dim (V\cap {\mathcal{C}}_{\mathrm{reg}})\le \dim V$.