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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)Hom_H(W,Res(V))=Hom_G(Ind(W),V) with HH a subgroup of GG 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)Ind(W) for a given representation WW of HH and how to prove the adjointness between ResRes and IndInd 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 2Vect2Vect 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 GG with values in 2Vect2Vect are, and can easily write a restriction functor ResRes from 2-representations of GG with values in 2Vect2Vect to 2-representations of a subgroup HH with values in 2Vect2Vect. Does ths have an adjoint IndInd? 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    * unpublished notes of mine on the induced 2-representation, where I tried to write it down from the groupoid-y "geometric" point of view, avoiding the choice of subgroups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!