Fixed-charge encoding and code projector

Fix a unitary fusion category $\mathcal{D}$ and the representative set $\mathrm{Irr}(\mathcal{D})$ of Multiplicity-aware fusion-tree basis. For an ordered list $x_1,\dots ,x_n\in \mathrm{Irr}(\mathcal{D})$, $n\ge 1$, a total charge $c\in \mathrm{Irr}(\mathcal{D})$, and a parenthesized input tensor $T$, let $V=\mathrm{Hom}(T,c)\mathrm{\ne }0$, equipped with that definition's orthonormal fusion-tree inner product. The identical-input case $x_1=\cdots =x_n=x$ retains the notation $V=V(x^n\to c)$. Let $\mathcal{C}={\mathbb{C}}^k$ with its ordered orthonormal computational basis, where $k\ge 1$. A fixed-charge encoding is an isometry $J:\mathcal{C}\to V$, so $J^{\ast }J=I_{\mathcal{C}}$. Its code subspace is $J\mathcal{C}$, and its code projector is $P:=J{J}^{\ast }:V\to V\mathrm{.}$ An encoding records a chosen subspace of a fusion space; it does not identify that subspace with an object of the fusion category.