TITLE: Non-symmetric Braiding on finite group Representation Categories
QUESTION [6 upvotes]: Do the fusion categories $Rep(S_4)$ and $Rep(A_5)$ admit non-symmetric braidings?  All the other rep. cats. of finite subgroups of $SU(2)$ do (in the McKay correspondence).  My guess is no.

REPLY [6 votes]: You can also answer the question using the classification of R-matrices over group algebras. Indeed, the R-matrices in the Hopf algebra $ kG $, are classified by pairs $ (A, \rho) $ where $ A \subset G $ is an abelian normal subgroup and $ \rho: A \times A \to k^* $ is a bilinear form ad $ G $-invariant. It is easy to see that the associated R-matrix is symmetric if and only if the bilinear form is skew-symmetric.
It is now clear that $ kA_5 $ does not have R-matrices in addition to the trivial. On the other hand, the only normal abelian subgroup of $ S_4 $ is formed by the union of the conjugacy classes of $(12)(34)$ and the identity of $S_4$. This normal subgroup is isomorphic to $C_2 \times C_2 $. As was mentioned by Ostrik, one example could be the next: the quadratic form that takes values -1 on nontrivial elements. That bilinear form satisfies the conditions and defines a non-symmetric R-matrix.