Are there two groups which are categorically Morita equivalent but only one of which is simple - MathOverflow

Are there two groups which are categorically Morita equivalent but only one of which is simple

2009-10-29T18:27:03Z

Can you find two finite groups G and H such that their representation categories are Morita equivalent (which is to say that there's an invertible bimodule category over these two monoidal categories) but where G is simple and H is not. The standard reference for module categories and related notions is this paper of Ostrik's

This is a much stronger condition than saying that C[G] and C[H] are Morita equivalent as rings (where C[A_7] and C[Z/9Z] gives an example, since they both have 9 matrix factors). It is weaker than asking whether a simple group can be isocategorical (i.e. have representation categories which are equivalent as tensor categories) with a non-simple group, which was shown to be impossible by Etingof and Gelaki.

Matt Emerton asked me this question when I was trying to explain to him why I was unhappy with any notion of "simple" for fusion categories. It's of interest to the study of fusion categories where the dual even Haagerup fusion category appears to be "simple" while the principal even Haagerup fusion category appears to be "not simple" yet the two are categorically Morita equivalent.

Answer by Noah Snyder for Are there two groups which are categorically Morita equivalent but only one of which is simple

2009-10-29T19:19:54Z

The farthest I got thinking about this problem (and I haven't thought about it all that much) is that module categories over C[G] are classified in Section 3.4. They correspond to pairs K a subgroup of G and a choice of central extension of K (or equivalently, a certain cohomology class). In the case where there's no central extension, the dual category is some sort of Hecke algebra category C[K\G/K]-mod that I've never totally understood. Also I don't know how to modify that construction when you introduce the central extension. Anyway, modulo understanding those issues the question comes down to when a twisted Hecke algebra category C[K\G/K]-mod can be equivalent as a tensor category to C[H]-mod for some group H.

Answer by pasquale zito for Are there two groups which are categorically Morita equivalent but only one of which is simple

2009-10-29T19:23:48Z

Wouldn't it follow that the quantum doubles of the two groups are isomorphic? Would this help to set the question? (Sorry for posting this as an answer, didn't manage to leave it as a comment).

Answer by David Jordan for Are there two groups which are categorically Morita equivalent but only one of which is simple

2009-10-31T15:42:31Z

Noah,

I think an answer to your question is given in http://arxiv.org/pdf/0810.0032, theorem 1.1.

Subcategories of the double D(G) are given by pairs of normal subgroups K,N in G which centralize each other, together with the data of a bicharacter K\times N \to C^\times.

So in particular if G has no normal subgroups and H does, then you're going to find that D(G) has no nontrivial subcategories, while D(H) will (one can take, K=the normal subgroup in H, N={id}, and the bicharacter K\to C^* to be trivial I guess.

-david

Answer by Dmitri Nikshych for Are there two groups which are categorically Morita equivalent but only one of which is simple

2010-01-05T06:11:14Z

Hi Noah,

Categorically Morita equivalent groups were studied by Deepak Naidu in arXiv:math/0605530. He obtained there a complete description of Morita equivalent groups. It is also shown that simple groups are categorically Morita rigid.

Best, Dmitri