# nLab pivotal symbols

The pivotal symbols or $3j$ symbols are a certain collection of signs attached to a fusion category arising from analyzing how invariant vectors in a $3$-particle state transform under rotation by a full revolution.

In a fusion category, the nature of duals for objects in the category can be encoded in principal ${ℂ}^{×}$-bundle over the simple objects. Firstly, suppose one goes ahead and chooses an arbitrary dual object ${V}_{i}^{*}$ for every simple object ${V}_{i}$. Write

(1)$\mathrm{Dual}\left({V}_{i}⊣{V}_{i}^{*}\right)$Dual(V_i \dashv V_i^*)

for the set of ways that ${V}_{i}^{*}$ can be expressed as a right dual of ${V}_{i}$. Notice that $\mathrm{Dual}\left({V}_{i}⊣{V}_{i}^{*}\right)$ is a ${ℂ}^{×}$-torsor (you can multiply the unit by a nonzero complex number, following which you are forced to multiply the counit by the reciprocal of this number if you want to obey the snake equations).

In a fusion category, if ${V}_{i}^{*}$ is a right dual of ${V}_{i}$, then it is also a left dual of ${V}_{i}$, though not in a canonical way. This gives us the principal ${ℂ}^{×}$-bundle $L$ over the simple objects — the fiber above ${V}_{i}$ is

(2)${L}_{{V}_{i}}=\left\{\mathrm{Dual}\left({V}_{i}⊣{V}_{i}^{*}\right)\stackrel{\sim }{\to }\mathrm{Dual}\left({V}_{i}^{*}⊣{V}_{i}\right)\right\},$L_{V_i} = \left\{Dual(V_i \dashv V_i^*) \stackrel{\sim}\rightarrow Dual(V_i^* \dashv V_i) \right\},

the set of isomorphisms between $\mathrm{Dual}\left({V}_{i}⊣{V}_{i}^{*}\right)$ and $\mathrm{Dual}\left({V}_{i}^{*}⊣{V}_{i}\right)$.

A pivotal structure or even-handed structure is a section of this bundle satisfying some naturality properties. In other words, it is a consistent way to turn right duals into left duals. The most stringent condition is that the invariant vectors in a tensor product of three particles must either be totally ‘bosonic’ or totally ‘fermionic’ when you rotate them through a full rotation.

That is, the endomorphism

(3)${T}_{\mathrm{ijk}}:\mathrm{Hom}\left(1,{V}_{i}\otimes {V}_{j}\otimes {V}_{k}\right)\to \mathrm{Hom}\left(1,{V}_{i}\otimes {V}_{j}\otimes {V}_{k}\right)$T_{ijk} : Hom(1,V_i \otimes V_j \otimes V_k) \rightarrow Hom(1,V_i \otimes V_j \otimes V_k)

(which can be proved to be an involution) given by sending

must be equal to $±\mathrm{id}$. Assuming this is satisfied (if not, there is no consistent way to turn right duals into left duals), we can call this collection of signs the $3j$ symbols or the pivotal symbols ${ϵ}_{\mathrm{ijk}}$ of the fusion category:

(4)${T}_{\mathrm{ijk}}={ϵ}_{\mathrm{ijk}}\mathrm{id}.$T_{ijk} = \epsilon_{ijk} id.

That is to say, the $3j$ symbols are the signs arising from choosing a trivialization of the ‘duality bundle’ and analyzing how invariant $3$-particle tensors transform when they go through a full revolution.

It turns out that one can express the concept of a pivotal structure directly in terms of the pivotal symbols ${ϵ}_{\mathrm{ijk}}$. Namely, a pivotal structure on the category amounts to an $ϵ$-twisted monoidal natural tranformation? of the identity on the category. That is to say, a collection of nonzero complex numbers ${t}_{i}$ satisfying

(5)${t}_{j}{t}_{k}={ϵ}_{\mathrm{ijk}}{t}_{i}$t_j t_k = \epsilon_{ijk} t_i

whenever ${V}_{i}$ appears in ${V}_{j}\otimes {V}_{k}$. If all the $3j$ symbols were equal to $1$, this would just be an ordinary monoidal natural transformation of the identity. Perhaps this phenomenon can be understood in a more enlightening way using the language of planar algebras.

# References

Revised on July 1, 2009 08:31:30 by Toby Bartels (71.104.230.172)