Timeline for Number of Skew Symmetric Matrices of fixed rank
Current License: CC BY-SA 3.0
9 events
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| Jul 10, 2016 at 22:14 | comment | added | Richard Stanley | It is known that the following three numbers are equal, but no combinatorial proof is known that any two are equal: (1) number of symmetric matrices in $\mathrm{GL}(2n,q)$ with zero diagonal, (2) number of symmetric matrices in $\mathrm{GL}(2n-1,q)$, (3) number of skew-symmetric matrices in $\mathrm{GL}(2n,q)$. See Enumerative Combinatorics, vol. 1, 2nd ed., Exercise 1.199. | |
| Jul 7, 2016 at 14:02 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jun 7, 2016 at 12:55 | answer | added | Friedrich Knop | timeline score: 2 | |
| Jun 7, 2016 at 11:30 | answer | added | Robert Bryant | timeline score: 0 | |
| Jun 6, 2016 at 18:11 | comment | added | Singh | Thank you for your comments. But I still have some doubts. I am not sure how to prove that the $GL(n,\mathbb{F})$ action is transitive. Could you please suggest me some reference. | |
| May 14, 2016 at 12:17 | comment | added | Robert Bryant | This problem is much easier than the symmetric case. As long as the characteristic of the field $\mathbb{F}$ is not $2$, the space $A_{n,r}$ of anti-symmetric $n$-by-$n$ matrices of rank $r$ (necessarily even) is a homogeneous space $\mathrm{GL}(n,\mathbb{F})/P_{n,r}$ where $P_{n,r}$ is the subgroup that stabilizes some particular element of $A_{n,r}$. Thus, you are essentially asking for the order of this subgroup, which is an easy exercise, so I expect one would find it as a remark in some paper about something else, rather than as a main result in something. Characteristic 2 may be harder. | |
| May 14, 2016 at 9:32 | history | edited | Denis Serre | CC BY-SA 3.0 |
added 4 characters in body; edited title
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| May 14, 2016 at 6:34 | review | First posts | |||
| May 14, 2016 at 9:01 | |||||
| May 14, 2016 at 6:28 | history | asked | Singh | CC BY-SA 3.0 |