F-symbol as a recoupling matrix

Let $a,b,c,d$ be simple charges in a unitary fusion category, and put $P_L=((a\otimes b)\otimes c)$ and $P_R=a\otimes (b\otimes c)$. The local fusion-tree basis result Fusion-tree basis and dimension recursion identifies

${\mathcal{H}}_L:={⨁}_e{V}_{ab}^e\otimes V_{ec}^d\text{with}\mathrm{Hom}(P_L,d),$

and

${\mathcal{H}}_R:={⨁}_f{V}_{bc}^f\otimes V_{af}^d\text{with}\mathrm{Hom}(P_R,d),$

by the corresponding tree compositions. These direct sums carry the orthonormal left- and right-tree bases. If $\mu :a\otimes b\to e$ and $\nu :e\otimes c\to d$ are vertex basis maps, write $L_{e,\mu ,\nu }:=\nu \circ (\mu \otimes {\mathrm{id}}_c):P_L\to d$. If $\rho :b\otimes c\to f$ and $\sigma :a\otimes f\to d$ are vertex basis maps, write $R_{f,\rho ,\sigma }:=\sigma \circ ({\mathrm{id}}_a\otimes \rho ):P_R\to d$. Let ${\alpha }_{a,b,c}:P_L\to P_R$ be the associator. Define the F-symbol matrix $F_d^{abc}$ by

$L_{e,\mu ,\nu }\circ {\alpha }_{a,b,c}^{-1}={\sum }_{f,\rho ,\sigma }{F}_d^{abc}[(f,\rho ,\sigma ),(e,\mu ,\nu )]R_{f,\rho ,\sigma }\mathrm{.}$

Rows index target-tree labels and multiplicity indices; columns index source-tree labels and multiplicity indices. The sums retain every fusion channel and every multiplicity coordinate. If the fixed-total fusion space is zero-dimensional, the F-symbol is the unique empty $0\times 0$ matrix.

The object-level associator is drawn below. The F matrix records its induced change of coordinates on the Hom spaces by the inverse-precomposition rule in the Definition; the diagram is only a convention aid.