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1. • CommentRowNumber1.
   • CommentAuthordomenico_fiorenza
   • CommentTimeMay 29th 2013
   • (edited May 29th 2013)
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexLet $H$ be a subgroup of a finite group $G$, and let $Res_H^G$ and $Ind_H^G$ the restriction and induction functors between the categories of linear representations of $G$ and $H$, respectively. I'm familiar with Frobenius reciprocity stating that $Ind_H^G$ is the right adjoint to $Res_H^G$; moreover, by choosing the traditional model for the induced representation \[ Ind_H^G(U)=\{f:G\to U such\,\, that f(h g)=h\cdot f(g)\,\, for\,\, any\,\, h\in H\} \] I also know how to write explicitly the natural isomorphism \[ Hom_{Rep(H)}(Res_H^G(W),U) \stackrel{\sim}{\to} Hom_{Rep(G)}(W,Ind_H^G(U)). \] Now, I've just learnt that actually $(Res_H^G,Ind_H^G)$ is an ambidextrous adjunction, i.e. that $Ind_H^G$ is also the left adjoint to $Res_H^G$, but I'm not familiar with this fact: how is the isomorphism \[ Hom_{Rep(H)}(U,Res_H^G(W)) \stackrel{\sim}{\to} Hom_{Rep(G)}(Ind_H^G(U),W) \] explicitly defined? (posting on MO, too. In case I will get an answer there I will report it here and vice versa)

   Let $H$ be a subgroup of a finite group $G$, and let $Res_H^G$ and $Ind_H^G$ the restriction and induction functors between the categories of linear representations of $G$ and $H$, respectively. I’m familiar with Frobenius reciprocity stating that $Ind_H^G$ is the right adjoint to $Res_H^G$; moreover, by choosing the traditional model for the induced representation

   $$Ind_H^G(U)=\{f:G\to U such\,\, that f(h g)=h\cdot f(g)\,\, for\,\, any\,\, h\in H\}$$

   I also know how to write explicitly the natural isomorphism

   $$Hom_{Rep(H)}(Res_H^G(W),U) \stackrel{\sim}{\to} Hom_{Rep(G)}(W,Ind_H^G(U)).$$

   Now, I’ve just learnt that actually $(Res_H^G,Ind_H^G)$ is an ambidextrous adjunction, i.e. that $Ind_H^G$ is also the left adjoint to $Res_H^G$, but I’m not familiar with this fact: how is the isomorphism

   $$Hom_{Rep(H)}(U,Res_H^G(W)) \stackrel{\sim}{\to} Hom_{Rep(G)}(Ind_H^G(U),W)$$

   explicitly defined?

   (posting on MO, too. In case I will get an answer there I will report it here and vice versa)

2. • CommentRowNumber2.
   • CommentAuthordomenico_fiorenza
   • CommentTimeMay 30th 2013
   • (edited May 30th 2013)
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexI got a complete answer by Qiaochu Yuan on <a href="http://mathoverflow.net/questions/132272/wrong-way-frobenius-reciprocity-for-finite-groups-representations/">MO</a>

   I got a complete answer by Qiaochu Yuan on MO

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 30th 2013
   • (edited May 30th 2013)
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   Author: Urs
   Format: MarkdownItexNice! I have now added a pointer to this discussion [here](http://ncatlab.org/nlab/show/induced+representation#ReferencesTraditionalFormulation). Would be nice if this were worked into the enty _[[induced representation]]_. Maybe you feel like doing so?

   Nice!

   I have now added a pointer to this discussion here. Would be nice if this were worked into the enty induced representation. Maybe you feel like doing so?

4. • CommentRowNumber4.
   • CommentAuthordomenico_fiorenza
   • CommentTimeJun 4th 2013
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexok.

   ok.