Timeline for How big can a wedge of $n$ 2-forms in $\mathbb{R}^{2n}$ be?
Current License: CC BY-SA 4.0
30 events
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| Aug 6, 2023 at 13:08 | comment | added | Mikhail Katz | @DeaneYang, I did. | |
| Aug 6, 2023 at 13:06 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
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| Jun 19, 2023 at 7:56 | answer | added | Denis Serre | timeline score: 5 | |
| Jun 18, 2023 at 16:14 | answer | added | Mikhail Katz | timeline score: 4 | |
| Jun 18, 2023 at 16:05 | history | edited | C.F.G | CC BY-SA 4.0 |
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| S Jun 18, 2023 at 9:21 | history | bounty ended | Mikhail Katz | ||
| S Jun 18, 2023 at 9:21 | history | notice removed | Mikhail Katz | ||
| Jun 18, 2023 at 9:21 | vote | accept | Mikhail Katz | ||
| Jun 17, 2023 at 1:55 | answer | added | Tom Goodwillie | timeline score: 14 | |
| Jun 16, 2023 at 14:16 | comment | added | Deane Yang | You mention in your answer to an earlier question that an answer to this leads to "natural extension of Gromov's optimal (i.e., tight) stable systolic inequality for the complex projective space". Any chance you want to elaborate on this a little (say, by editing your question)? | |
| Jun 16, 2023 at 10:42 | comment | added | David E Speyer | Yes, that's correct. | |
| Jun 16, 2023 at 9:18 | comment | added | Mikhail Katz | @DavidESpeyer, so is it correct that one can reformulate the problem in a comass-free form as follows? Let $P(*,*,\ldots,*)$ be the unique symmetric multilinear polynomial which, along the diagonal $(A,\ldots,A)$, specializes to $n!$ times the Pfaffian of $A$. Now let $A_1,\ldots,A_n$ be antisymmetric matrices of unit spectral radius. Then the value of $P$ on this $n$-tuple is also at most $n!$. ?? | |
| Jun 16, 2023 at 7:38 | comment | added | Mikhail Katz | @DanielAsimov: Good, I am convinced. Can you provide a proof, or else as Herman to provide one one? | |
| Jun 15, 2023 at 21:59 | comment | added | Daniel Asimov | I will be utterly astonished if there is a larger maximum than the n! as in the question. | |
| Jun 15, 2023 at 18:16 | history | edited | Michael Hardy | CC BY-SA 4.0 |
The conspicuous lack of proper horizontal spacing between "n!" and "Pfaffian" resulted from the use of \text{} where \operatorname{} should be used. The latter has context-dependent spacing.
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| S Jun 15, 2023 at 18:09 | history | bounty started | Mikhail Katz | ||
| S Jun 15, 2023 at 18:09 | history | notice added | Mikhail Katz | Draw attention | |
| Jun 15, 2023 at 12:18 | comment | added | David E Speyer | I haven't been able to find a direct implication. The trouble is that the relationship $\text{Pfaffian}^2 = \text{Det}$ turns into a much messier relationship between $P$ and $D$. | |
| Jun 15, 2023 at 9:03 | comment | added | Mikhail Katz | @DavidESpeyer: Is the $D(A,\ldots,A)$ question only a variant of the 2-form question, or is there an implication between them? | |
| Jun 14, 2023 at 16:17 | comment | added | Mikhail Katz | @DavidESpeyer, Thanks for all your help! Note that my remark in the original version of the question about a bound for a foursome of 2-forms (which is not as good as $4!$ but much better than existing bounds for the general case) seems to have been lost in the shuffle. | |
| Jun 13, 2023 at 14:59 | comment | added | David E Speyer | I'll make one more comment. Here is a variant of the problem which doesn't mention skew-symmetry and which might already be known. Let $D(A^1, A^2, \ldots, A^m)$ be the unique symmetric multilinear polynomial in $m$ many $m \times m$ matrices such that $D(A, A, \dots, A) = \det(A)$. Suppose that each $A^k$ has operator norm $\leq 1$. Can we conclude that $|D(A^1, A^2, \ldots, A^m)| \leq 1$? | |
| Jun 13, 2023 at 14:55 | history | edited | David E Speyer | CC BY-SA 4.0 |
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| Jun 13, 2023 at 14:39 | history | edited | David E Speyer | CC BY-SA 4.0 |
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| Jun 13, 2023 at 14:27 | history | edited | David E Speyer | CC BY-SA 4.0 |
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| Jun 13, 2023 at 14:20 | comment | added | Mikhail Katz | Thanks a lot! @DavidESpeyer | |
| Jun 13, 2023 at 14:18 | comment | added | David E Speyer | Okay, edits made. As you can tell, I like this question! | |
| Jun 13, 2023 at 14:17 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 2174 characters in body; edited tags; edited title
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| Jun 13, 2023 at 14:00 | comment | added | Mikhail Katz | @DavidESpeyer I would much appreciate it, go ahead. | |
| Jun 13, 2023 at 13:59 | comment | added | David E Speyer | Would you mind if I edited this to include the background from the previous questions? While at it, I'd also really like to tweak the notation in two ways: (1) Ask about $n$ two-forms, so as not to suggest an infinite sequence of questions :) and (2) use the notation $\alpha_1$, $\alpha_2$, ..., $\alpha_n$ rather than $\alpha$, $\beta$, $\gamma$, ... since the alphabetic notation is a pain to engage with. (EG: Imagine writing "let the eigenvalues of $\alpha$ be $(\rho_1, \rho_2, \dots, \rho_{2n})$, the eigenvalues of $\beta$ be $(\sigma_1, \sigma_2, \dots, \sigma_{2n})$ etcetera ...") | |
| Jun 13, 2023 at 13:32 | history | asked | Mikhail Katz | CC BY-SA 4.0 |