Multiplicity-aware fusion-tree basis

Let $\mathcal{C}$ be a unitary fusion category and fix a set $\mathrm{Irr}(\mathcal{C})$ of representatives for the simple isomorphism classes, with the tensor unit $\mathbf{1}$ itself representing its class. Throughout this definition, the ordered input charges $a_1,\dots ,a_n$, total charge $d$, and all internal edge charges are labels in $\mathrm{Irr}(\mathcal{C})$. Also fix a parenthesized binary fusion tree. At each internal vertex with incoming charges $x,y$ and outgoing charge $z$, choose one vector from an orthonormal basis of $V_{xy}^z$. An admissible labeling assigns a representative charge to each internal edge so that each chosen vertex basis is nonempty; a separate multiplicity-basis index is retained at every vertex. Compose the vertex morphisms, inserting the associators required by the chosen tree, to obtain a morphism from the parenthesized input tensor to $d$. These composite morphisms are the fusion-tree vectors. The family is called a fusion-tree basis when it is an orthonormal basis of the fixed-input, fixed-total fusion space. For the empty input list the tensor is $\mathbf{1}$ and its empty tree vector is ${\mathrm{id}}_{\mathbf{1}}$ when $d=\mathbf{1}$; for one input $a_1$, the only tree vector is ${\mathrm{id}}_{a_1}$ when $d=a_1$. The next local lemma proves the basis property for every finite input list and includes the dimension recursion (Fusion-tree basis and dimension recursion ↗).

The left-associated three-input tree has leaves $a,b,c$ in that order, internal charge $e$ after fusing $a,b$, and output $d$ after fusing $e,c$:

The drawing records only the parenthesization and edge labels. The basis vector is the composite morphism specified above; no proof depends on the diagram.