Question

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.

Answer 1

Eric,

The answer is no for $Rep(A_5)$ and yes for $Rep(S_4)$, thanks to Victor Ostrik's observation. For a braided category $C$ let $C'$ denote its Mueger center, i.e., the subcategory of objects $Y$ in $C$ such that the square of braiding of $Y$ with any $X$ in $C$ is identity. So $C$ is symmetric if $C=C'$ and $C$ is non-degenerate (or modular) if $C'$ is trivial.

Note that $C:=Rep(A_5)$ is simple, i.e., it has no non-trivial proper fusion subcategories. Now if $C$ has a non-symmetric braiding then $C' \neq C$ is a proper subcategory. So $C'$ is trivial, i.e., $C$ with the above braiding is non-degenerate (modular). This cannot happen (e.g., $C$ has a simple object of dimension 5, but in a modular category the square of dimension of any object divides dimension of the category, thanks to the result of Etingof-Gelaki).

For $D:= Rep(S_4)$ there is a non-symmetric braiding with $D'=Rep(S_3)$, namely the equivariantization of a pointed category $Vec_{Z/2Z\oplus Z/2Z}$ with respect to an action of $S_3$.

Answer 2

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.