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1. At Frobenius reciprocity I can see Frobenius reciprocity can be formulated in very general terms and that the classical adjoint pair in the theory of finite dimensional representations of finite groups $Hom_H(W,Res(V))=Hom_G(Ind(W),V)$ with $H$ a subgroup of $G$ is just a very particular case of the general theory. However, this familiar case has (to me) the advantage that I know how to explicitly compute $Ind(W)$ for a given representation $W$ of $H$ and how to prove the adjointness between $Res$ and $Ind$ by hand. So I’m wondering about what can be said about linear 2-representations of finite groups (with, I guess, some finiteness assumptions I’m not able to specify at the moment). For instance, if we take as a model for $2Vect$ the 2-category algebras/bimodules/bimodule morphisms then I have a clear idea of what the explcit data of a representation of a finite group $G$ with values in $2Vect$ are, and can easily write a restriction functor $Res$ from 2-representations of $G$ with values in $2Vect$ to 2-representations of a subgroup $H$ with values in $2Vect$. Does ths have an adjoint $Ind$? How is this explicitely described? (I should be able to work this out by myself by thinking to it enough, but I would like not to loose time on this if it is already well known and a pointer to the literature will solve this)

• CommentRowNumber2.
• CommentAuthorBruce Bartlett
• CommentTimeNov 7th 2012
Hi Domenico,

For 2-representations of finite groups in the not-the-same-but-related "KV" model of 2Vect where objects are semisimple linear categories, morphisms are linear functors, 2-morphisms are natural transformations, here are some references for induction:

* Ganter and Kapranov's paper describes it explicitly.
* This paper by Anjelica Osorno identifies the induced representation in terms of cohomological data using the Shapiro isomorphism.
* unpublished notes of mine on the induced 2-representation, where I tried to write it down from the groupoid-y "geometric" point of view, avoiding the choice of subgroups and stabilizers etc, see "Functoriality for the 2-character of 2-representations of groups via even-handed structures" at my webpage.
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 7th 2012
• (edited Nov 7th 2012)

the not-the-same-but-related “KV” model of 2Vect

It’s a sub-model. KV 2-vector spaces form a sub-2-category of those that Domenico mentioned.

• CommentRowNumber4.
• CommentAuthorBruce Bartlett
• CommentTimeNov 8th 2012
Yes, good point, it's a sub-model.
2. Hi Bruce,

Thanks a lot!