Timeline for Wrong-way Frobenius reciprocity for finite groups representations
Current License: CC BY-SA 3.0
15 events
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| S Sep 8, 2016 at 15:17 | history | suggested | CommunityBot | CC BY-SA 3.0 |
replaced several \_ by _ (formating seems more natural this way)
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| Sep 8, 2016 at 13:51 | review | Suggested edits | |||
| S Sep 8, 2016 at 15:17 | |||||
| May 30, 2013 at 17:51 | comment | added | Theo Johnson-Freyd | ... I believe that this $\mathbb G_m$ action is precisely the $\mathrm{SO}(2)$ action guaranteed by results of Lurie. Note that $\mathrm{SO}(2) \sim S^1$ has representation theory $\mathrm{Vect}[L]$, where $L$ is a line; but if you perform Tannakian reconstruction on this category, you get back the affine algebraic group $\mathbb G_m$. | |
| May 30, 2013 at 17:47 | comment | added | Theo Johnson-Freyd | Incidentally, I agree that there is one "most natural" isomorphism between the induction and coinduction functors, but it's of course not the only one: you may multiply by any element of $\mathbb G_m = \mathbb C^\times$. In general, the space of left adjoints to some functor has the homotopy type of a truth value (it is either empty or contractible), and as is the space of right adjoints, but the space of biadjoints is in general the intersection of two contractible sets, and so can have arbitrary homotopy type. (For functors between $1$-categories, it is never worse than a set.) ... | |
| May 30, 2013 at 7:59 | comment | added | Qiaochu Yuan | @domenico: yes, but perhaps it's cleaner to start from the map in the other direction. | |
| May 30, 2013 at 7:20 | comment | added | domenico fiorenza | Qiaochu Yuan, thanks a lot for the explicit map you added to your answer. Now everything is crystal clear, thanks! To be sure I'm correctly interpreting it, you are secretely identifying $\mathbb{C}[G]\otimes_{\mathbb{C}[H]}V$ with the subspace of $\mathbb{C}[G]\otimes_{\mathbb{C}}V$ consisting of those $\sum_{g\in G} g\otimes v_g$ such that $v_{gh}=h^{-1}v_g$ for any $h\in H$, via the natural projection $\mathbb{C}[G]\otimes_{\mathbb{C}}V\to \mathbb{C}[G]\otimes_{\mathbb{C}[H]}V$, whose inverse is $g\otimes_{\mathbb{C}[H]}v\mapsto \frac{1}{|H|}\sum_{h\in H}gh^{-1}\otimes hv$, right? | |
| May 30, 2013 at 5:15 | vote | accept | domenico fiorenza | ||
| May 30, 2013 at 2:43 | comment | added | user28172 | Can't one just use that restriction commutes with duality and consider the dual of the induction of the dual? Or more conceptually, for general groups $G$ and $H$ there is induction and there is "compactly supported" induction, the latter contained in the former and equality when $H$ has finite index in $G$. In general these functors have opposite adjointness properties with respect to restriction (much like direct sum versus direct product), so when the two functors agree one sees this common functor having two adjointeness properties relative to restriction. | |
| May 29, 2013 at 21:10 | comment | added | Qiaochu Yuan | I added an explicit map. Something went wrong when I tried to write this map down abstractly and I'm not sure how to fix it. | |
| May 29, 2013 at 21:09 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| May 29, 2013 at 21:03 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| May 29, 2013 at 20:39 | comment | added | domenico fiorenza | Thanks! a really enlightening answer, putting the problem into its correct abstract setting! Yet, to complete a few computations I'm currently fighting with, what I'd need is a completely explicit version of the isomorphism $Hom_{Rep(H)}(U,Res_H^G(W)) \stackrel{\sim}{\to} Hom_{Rep(G)}(Ind_H^G(U),W)$, with the explicit model for the induced representation mentioned above. I think I can work this out from the answer, but I'll wait still a couple of days to see if there's someone knowing this on the spot (and willing to post it here :) ) | |
| May 29, 2013 at 20:31 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| May 29, 2013 at 20:25 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| May 29, 2013 at 20:18 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |