We all learned in kindergarten that the category of finite-dimensional (type I, say) $U_q(\mathfrak{g})$-modules is braided monoidal for $\mathfrak{g}$ a complex semisimple Lie algebra. This gives an action of the braid group $B_n$ on $V^{\otimes n}$ for any object $V$ in the category.

In order to construct certain maps on $V^{\otimes n}$, e.g. $q$-analogues of symmetrizers, antisymmetrizers, there is a problem: classically, the symmetrizer is defined as a sum over permutations, whereas in this case we have only an action of the braid group, not the symmetric group. To resolve this, one constructs a nice splitting of the quotient map $B_n \to S_n$ as follows: write a permutation as a minimal product of adjacent transpositions, and then replace each transposition with the braiding. Along the way you have to prove that any two such minimal decompositions of a permutation can be transformed into one another using only the braid relations; this proves that the splitting map is well-defined. Note that this map isn't a homomorphism, but it has some nice properties.

For various reasons, it is sometimes nice to consider a modification of the braiding. Drinfeld calls this a unitarized R-matrix, while some others (e.g. Kamnitzer, Henriques, Tingley) call this the cactus commutor. This guy squares to the identity but does not satisfy the braid relations, although there is something else called the cactus relation or the coboundary relation. This gives an action of the cactus group $J_n$ on $V^{\otimes n}$.

**Question: is there an analogous splitting of the quotient map $J_n \to S_n$?**