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1. • CommentRowNumber1.
   • CommentAuthordomenico_fiorenza
   • CommentTimeOct 24th 2012
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   Author: domenico_fiorenza
   Format: MarkdownItexAt [[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)

   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)

2. • CommentRowNumber2.
   • CommentAuthorBruce Bartlett
   • CommentTimeNov 7th 2012
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   Author: Bruce Bartlett
   Format: HtmlHi 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 <a href="http://arxiv.org/abs/math/0602510">paper</a> describes it explicitly. * <a href="http://arxiv.org/abs/0904.4414">This paper</a> 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 <a href="http://math.sun.ac.za/~bbartlett/research/">my webpage</a>.

   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.
3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 7th 2012
   • (edited Nov 7th 2012)
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   Author: Urs
   Format: MarkdownItex> 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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorBruce Bartlett
   • CommentTimeNov 8th 2012
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   Author: Bruce Bartlett
   Format: HtmlYes, good point, it's a sub-model.

   Yes, good point, it's a sub-model.
5. • CommentRowNumber5.
   • CommentAuthordomenico_fiorenza
   • CommentTimeNov 22nd 2012
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   Author: domenico_fiorenza
   Format: MarkdownItexHi Bruce, Thanks a lot!

   Hi Bruce,

   Thanks a lot!