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Atrium > Mathematics, Physics & Philosophy: Wrong-way Frobenius reciprocity for finite groups representations

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- 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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>$ be a subgroup of a finite group $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>$ , and let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>Res</mi> <mi>H</mi> <mi>G</mi></msubsup></mrow><annotation encoding="application/x-tex">Res_H^G</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>Ind</mi> <mi>H</mi> <mi>G</mi></msubsup></mrow><annotation encoding="application/x-tex">Ind_H^G</annotation></semantics></math>$ the restriction and induction functors between the categories of linear representations of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>$ , respectively. I’m familiar with Frobenius reciprocity stating that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>Ind</mi> <mi>H</mi> <mi>G</mi></msubsup></mrow><annotation encoding="application/x-tex">Ind_H^G</annotation></semantics></math>$ is the right adjoint to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>Res</mi> <mi>H</mi> <mi>G</mi></msubsup></mrow><annotation encoding="application/x-tex">Res_H^G</annotation></semantics></math>$ ; moreover, by choosing the traditional model for the induced representation
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>Ind</mi> <mi>H</mi> <mi>G</mi></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>f</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>U</mi><mi>such</mi><mspace width="0.16667em"/><mspace width="0.16667em"/><mi>that</mi><mi>f</mi><mo stretchy="false">(</mo><mi>h</mi><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi>h</mi><mo>⋅</mo><mi>f</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.16667em"/><mspace width="0.16667em"/><mi>for</mi><mspace width="0.16667em"/><mspace width="0.16667em"/><mi>any</mi><mspace width="0.16667em"/><mspace width="0.16667em"/><mi>h</mi><mo>∈</mo><mi>H</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> Ind_H^G(U)=\{f:G\to U such\,\, that f(h g)=h\cdot f(g)\,\, for\,\, any\,\, h\in H\} </annotation></semantics></math>$
I also know how to write explicitly the natural isomorphism
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>Hom</mi> <mrow><mi>Rep</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>Res</mi> <mi>H</mi> <mi>G</mi></msubsup><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo><mo>,</mo><mi>U</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>∼</mo></mover><msub><mi>Hom</mi> <mrow><mi>Rep</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><msubsup><mi>Ind</mi> <mi>H</mi> <mi>G</mi></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{Rep(H)}(Res_H^G(W),U) \stackrel{\sim}{\to} Hom_{Rep(G)}(W,Ind_H^G(U)). </annotation></semantics></math>$
Now, I’ve just learnt that actually $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Res</mi> <mi>H</mi> <mi>G</mi></msubsup><mo>,</mo><msubsup><mi>Ind</mi> <mi>H</mi> <mi>G</mi></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Res_H^G,Ind_H^G)</annotation></semantics></math>$ is an ambidextrous adjunction, i.e. that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>Ind</mi> <mi>H</mi> <mi>G</mi></msubsup></mrow><annotation encoding="application/x-tex">Ind_H^G</annotation></semantics></math>$ is also the left adjoint to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>Res</mi> <mi>H</mi> <mi>G</mi></msubsup></mrow><annotation encoding="application/x-tex">Res_H^G</annotation></semantics></math>$ , but I’m not familiar with this fact: how is the isomorphism
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>Hom</mi> <mrow><mi>Rep</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msubsup><mi>Res</mi> <mi>H</mi> <mi>G</mi></msubsup><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mo>∼</mo></mover><msub><mi>Hom</mi> <mrow><mi>Rep</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>Ind</mi> <mi>H</mi> <mi>G</mi></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom_{Rep(H)}(U,Res_H^G(W)) \stackrel{\sim}{\to} Hom_{Rep(G)}(Ind_H^G(U),W) </annotation></semantics></math>$
explicitly defined?

(posting on MO, too. In case I will get an answer there I will report it here and vice versa)
- 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
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeMay 30th 2013
- (edited May 30th 2013)
- PermaLink
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?
- CommentRowNumber4.
- CommentAuthordomenico_fiorenza
- CommentTimeJun 4th 2013
- PermaLink
Author: domenico_fiorenza
Format: MarkdownItexok.

ok.

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Atrium > Mathematics, Physics & Philosophy: Wrong-way Frobenius reciprocity for finite groups representations