Say I have two finite groups $G$ and $H$ which aren't isomorphic but have the same character table (for example, the quaternion group and the symmetries of the square). Does this mean that the corresponding categories of finite dimensional complex representations are isomorphic (ignoring the forgetful functor to vector spaces), or just that the corresponding representation rings are?



Finite groups with the same character table
In the particular case of the non-abelian groups of order 8, their categories of modules are not equivalent as monoidal categories. That they're not equivalent as pivotal categories can be proved by looking at the Frobenius-Schur indicator (I learned this from a paper of Susan Montgomery). That they're not equivalent even as monoidal categories can be proved by counting the fiber functors to vector spaces and seeing that one has more in one case (I can't remember which paper I saw this in, but almost surely Pavel Etingof was one of the coauthors).


