Timeline for Finite, abelian, yet "fugitive" orthogonal subgroups
Current License: CC BY-SA 3.0
26 events
| when toggle format | what | by | license | comment | |
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| S Mar 6, 2016 at 17:09 | history | suggested | LSpice | CC BY-SA 3.0 |
Added links; slightly improved impromptu eqnarray (I think)
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| Mar 6, 2016 at 16:49 | review | Suggested edits | |||
| S Mar 6, 2016 at 17:09 | |||||
| S Mar 5, 2016 at 16:02 | history | edited | Tony Pantev | CC BY-SA 3.0 |
fixed LaTeX typo that was preventing the definition of $\chi_g$ from parsing.
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| S Mar 5, 2016 at 16:02 | history | suggested | benblumsmith | CC BY-SA 3.0 |
fixed LaTeX typo that was preventing the definition of $\chi_g$ from parsing. Also, changed "allows to quickly derive" to "allows one to quickly derive", just to meet the requirement that an edit be at least 6 characters; hopefully this change is innocuous.
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| Mar 5, 2016 at 15:56 | review | Suggested edits | |||
| S Mar 5, 2016 at 16:02 | |||||
| Jul 30, 2013 at 9:24 | comment | added | Juan Bermejo Vega | @Emerton, thanks for your comments! | |
| Jul 30, 2013 at 4:48 | comment | added | Emerton | ... properties that I used here (in particular, the exact sequence $$0 \to H\cap K \to H \times K \to G$$ converted to an exact sequence in the other direction once I took duals), and property (4) is a basic consequence of them. Unfortunately, texts don't always list all the basic consequences of the properties that they state ... . Regards, Matthew | |
| Jul 30, 2013 at 4:45 | comment | added | Emerton | Dear Juan, Here is the standard way to deduce (4): Consider the homomorphism $H \times K \to G$ given by $(h,k) \mapsto h - k$. The kernel is precisely $H \cap K$ (embedded diagonally inside $H \times K$). Passing to duals, obtain a right exact sequence $$\hat{G} \to \hat{H} \times \hat{K} \to \widehat{H\cap K} \to 0,$$ which we may rewrite as $$\hat{G} \to \hat{G}/H^{\perp} \times \hat{G}/K^{\perp} \to \hat{G}/(H\cap K)^{\perp} \to 0.$$ This gives $(H\cap K)^{\perp} = \langle H^{\perp}, K^{\perp}\rangle $. Many texts on character theory for abelian groups will contain the basic exactness | |
| Jul 30, 2013 at 1:52 | answer | added | Francois Ziegler | timeline score: 5 | |
| Jul 29, 2013 at 11:53 | history | edited | Juan Bermejo Vega | CC BY-SA 3.0 |
updated question after accepting answer
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| Jul 29, 2013 at 9:18 | vote | accept | Juan Bermejo Vega | ||
| Apr 20, 2012 at 17:34 | answer | added | Simon Lentner | timeline score: 1 | |
| Feb 28, 2012 at 11:28 | history | edited | Juan Bermejo Vega | CC BY-SA 3.0 |
improved format; added 3 characters in body
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| Feb 28, 2012 at 9:01 | comment | added | Marc Palm | en.wikipedia.org/wiki/Character_theory#Orthogonality_relations | |
| Feb 28, 2012 at 8:49 | comment | added | Juan Bermejo Vega | @Geoff. I agree, that's part of what I tried to say. The issues are, in quantum physicis (my field) most readers would not be familiar with character theory. Thus, in a paper I prefer to work out these properties or to cite a clear reference. In this case, this concept of "orhogonal groups" is very popular in Quantum Computation and has been used quite extensively in algorithm design for more than a decade. Therefore, I am still interested on knowing what is the most common name for it in Math, and on finding it somewhere on the standard literature. | |
| Feb 27, 2012 at 18:21 | comment | added | Marc Palm | Perhaps the key word is not orthogonal subgroups, but lattices and dual lattices. | |
| Feb 27, 2012 at 17:58 | history | edited | Juan Bermejo Vega | CC BY-SA 3.0 |
improved format; added 7 characters in body; added 29 characters in body
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| Feb 27, 2012 at 16:02 | answer | added | BS. | timeline score: 9 | |
| Feb 27, 2012 at 15:44 | comment | added | Geoff Robinson | These are quite easy facts which anyone who knows about the rudiments of character theory of finite groups could prove in a few minutes. I coud not say when they were first explictly noted, but it is just a question of writing down the definitions, so I would be surprised if much older references do not exist. | |
| Feb 27, 2012 at 15:16 | comment | added | Juan Bermejo Vega | Actually, if I choose $G=\mathbf{Z}_d^m$, then it seems that the "orthogonal subgroup" is the syzygy of a module: ams.org/notices/200604/what-is.pdf | |
| Feb 27, 2012 at 14:48 | comment | added | Juan Bermejo Vega | I clarified the notation for $\chi_g$, typically in our field one starts with a cyclic decomposition of $G$ and then the notation makes more sense. Is it clear now? Also, I removed the join entirely, since one can say the same thing using generating brackets, which are more common. | |
| Feb 27, 2012 at 14:45 | history | edited | Juan Bermejo Vega | CC BY-SA 3.0 |
clarified notation
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| Feb 27, 2012 at 14:21 | history | edited | Juan Bermejo Vega | CC BY-SA 3.0 |
improved format
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| Feb 27, 2012 at 14:09 | comment | added | darij grinberg | What is $\chi_g$? Also, you can write $+$ for the join, since the groups are abelian and thus the join is just the sum of $\mathbb Z$-submodules. | |
| Feb 27, 2012 at 14:03 | history | edited | Juan Bermejo Vega | CC BY-SA 3.0 |
improved format
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| Feb 27, 2012 at 13:57 | history | asked | Juan Bermejo Vega | CC BY-SA 3.0 |