Unitary fusion categories (UFCs) have gained increased attention due to emerging connections with quantum physics. We consider a fusion rule of the form $q\otimes q \cong \mathbf{ 1}\oplus\bigoplus^k_{ i=1}x_{ i}$ in a UFC $\mathcal{ C}$, and extract information using the graphical calculus. For instance, we classify all associated skein relations when $k=1,2$ and $\mathcal{ C}$ is ribbon. In particular, we also consider the instances where $q$ is antisymmetrically self-dual. Our main results follow from considering the action of a rotation operator on a "canonical basis". Assuming self-duality of the summands $x_{ i}$, some general observations are made e.g. the real-symmetricity of the $F$-matrix $F^{ qqq}_q$. We then find explicit formulae for $F^{ qqq}_q$ when $k=2$ and $\mathcal{ C}$ is ribbon, and see that the spectrum of the rotation operator distinguishes between the Kauffman and Dubrovnik polynomials.
Categories: math.QA math-ph math.GT math.MP math.RT quant-ph