Timeline for Non-degenerate alternating bilinear form on a finite abelian group
Current License: CC BY-SA 3.0
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Aug 14, 2015 at 5:46 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Jul 29, 2011 at 8:17 | comment | added | Tobias Kildetoft | Also, it might be noted (since this is not obvious from the notation) that in fact the group of characters of $G$ is isomorphic to $G$ (since $G$ is abelian), so even though it at first glance just looks like we get that the order of $A$ is square, we actually get that in some sense $A$ itself is square. | |
| Mar 21, 2011 at 10:40 | vote | accept | Giuseppe | ||
| Mar 21, 2011 at 10:39 | comment | added | Giuseppe | Thank you for the link Francesco; I am now convinced that this proof is very natural, and I'll accept your answer! | |
| Mar 20, 2011 at 0:12 | comment | added | Amritanshu Prasad | This result comes up in the context of classifying Heisenberg central extensions of finite abelian groups. We discuss this aspect in the intro to our paper Locally Compact Abelian Groups with Symplectic Seld-duality, by Prasad, Shapiro and Vemuri (Adv. Math. 225 (2010) 2429-2454), arxiv.org/abs/0906.4397. It also seems to come up quite naturally in the theory of abelian varieties. It continues to hold for most reasonable classes of locally compact abelian groups, but fails in general. | |
| Mar 19, 2011 at 19:33 | comment | added | Pete L. Clark | To me this result has a folkloric feel: I am certainly aware of it and have used it in my own work (I even use the phrase "Lagrangian decomposition" in this context, which I think I picked up from some notes of van der Geer and Moonen), at least since 2003. But I don't know where this result comes from originally: does anyone have a primary source? | |
| Mar 18, 2011 at 15:46 | comment | added | Francesco Polizzi | I added in the answer a remark pointing out the analogy with symplectic spaces | |
| Mar 18, 2011 at 15:38 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
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| Mar 18, 2011 at 15:24 | comment | added | Francesco Polizzi | This proof is very natural because it really rephrases, in the contest of finite groups, the usual proof for symplectic vector spaces. In fact, any finite-dimensional symplectic vector space $V$ can be written as $W \oplus W^*$, where $W$ is a lagrangian (=isotropic of maximal dimension) subspace for the given symplectic form. This explains why $\dim V$ must be even. Look at en.wikipedia.org/wiki/Symplectic_vector_space, in particular the paragraph "Standard symplectic space" | |
| Mar 18, 2011 at 14:01 | comment | added | Giuseppe | That proof is quite swish, thank you. I still feel though that it should follow from the analogous result from linear algebra for vector spaces. | |
| Mar 18, 2011 at 12:08 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
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| Mar 18, 2011 at 12:00 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
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| Mar 18, 2011 at 11:54 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
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| Mar 18, 2011 at 11:47 | history | answered | Francesco Polizzi | CC BY-SA 2.5 |