Timeline for A slick proof of the Bruhat Decomposition for GL_n(k)?
Current License: CC BY-SA 2.5
31 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 15, 2013 at 18:41 | answer | added | ACL | timeline score: 17 | |
| May 2, 2012 at 17:25 | answer | added | Amritanshu Prasad | timeline score: 19 | |
| Jun 23, 2010 at 19:54 | comment | added | Harry Gindi | Alright, fair. You win,sir! | |
| Jun 23, 2010 at 19:48 | comment | added | Ryan Reich | Is this short enough: math.harvard.edu/~ryanr/bruhat_row-reduction.pdf? | |
| Jun 23, 2010 at 16:55 | comment | added | Harry Gindi | Emerton explained everything in full. If he were writing down the proof without all of the exposition, it would be significantly shorter, yes. | |
| Jun 23, 2010 at 16:47 | comment | added | Ryan Reich | Longer than Emerton's (accepted) answer? | |
| Jun 23, 2010 at 8:00 | comment | added | Harry Gindi | I don't understand what we're discussing here! The row-reduction proof is a proof by induction. If you in fact try to write it out, it is fairly long. | |
| Jun 23, 2010 at 0:55 | comment | added | Ryan Reich | Row reduction can give you the disjoint union easily. It is not generalizable, but it is not an inferior proof. | |
| Jun 22, 2010 at 17:10 | comment | added | Harry Gindi | @Victor, yes, of course. That's why I wrote the disjoint union rather than the union. | |
| Jun 22, 2010 at 17:09 | comment | added | Harry Gindi | "We share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury." - Irving Kaplansky about himself and Paul Halmos. | |
| Jun 22, 2010 at 9:36 | comment | added | GS | I don't see any reason to beat up on dependable ol' row-reduction. I seem to remember a quote by a famous algebraist: "I think coordinate-free, I write coordinate-free, but when the chips are down I lock the door and compute like hell with matrices." | |
| Jun 22, 2010 at 5:15 | comment | added | Victor Protsak | Also, Bruhat decomposition says more than $G=BWB,$ namely, that the cells $BwB$ are disjoint for different $w\in W.$ This fact is rather important. | |
| Jun 22, 2010 at 5:13 | comment | added | Victor Protsak | A warning: although in the case of the reductive algebraic group $G=GL_n,$ the Weyl group $W=N(T)/T\simeq S_n$ can be realized as a subgroup of $G$, for the simple algebraic group $SL_n,$ $W\simeq S_n$ cannot be realized as a subgroup of $G.$ So when we talk about Bruhat decomposition in semisimple algebraic groups, elements $w$ by themselves don't make sense, but the cells $BwB$ do (check that they are independent of the choice of coset representatives!). | |
| Jun 21, 2010 at 20:34 | comment | added | Harry Gindi | Yes, that is true, but proving it that way is a nasty proof by induction. The proof I did was slicker than that. | |
| Jun 21, 2010 at 19:46 | comment | added | Ryan Reich | This is very coordinate independent, but you should know that the Bruhat decomposition for $GL_n(k)$ is just "row reduction" from the first week of linear algebra. The rightmost $B$ is the "echelon form" of a matrix, the factor of $w$ accounts for rows having to be reordered, and the leftmost $B$ is the coefficient matrix of the Gauss-Jordan elimination algorithm. | |
| Feb 17, 2010 at 7:27 | comment | added | Kim Morrison | I have just removed some off-topic discussion. The original "content" is preserved on meta: tea.mathoverflow.net/discussion/219/… | |
| Feb 17, 2010 at 7:26 | history | rollback | Kim Morrison |
Rollback to Revision 3
|
|
| Feb 17, 2010 at 5:54 | history | edited | Harry Gindi | CC BY-SA 2.5 |
added 314 characters in body; added 269 characters in body
|
| Feb 17, 2010 at 5:49 | history | edited | Harry Gindi | CC BY-SA 2.5 |
added 852 characters in body
|
| Feb 17, 2010 at 4:17 | vote | accept | Harry Gindi | ||
| Feb 17, 2010 at 4:12 | answer | added | Emerton | timeline score: 57 | |
| Feb 17, 2010 at 4:02 | history | edited | Peter McNamara |
edited tags
|
|
| Feb 17, 2010 at 3:19 | answer | added | Allen Knutson | timeline score: 17 | |
| Feb 17, 2010 at 3:15 | comment | added | Faisal | You can check out section 23.4 of Fulton and Harris for a direct proof for SL(V). | |
| Feb 17, 2010 at 2:52 | answer | added | Peter McNamara | timeline score: 2 | |
| Feb 17, 2010 at 2:50 | answer | added | Tom Church | timeline score: 13 | |
| Feb 17, 2010 at 2:42 | comment | added | Shizhuo Zhang | I think one can use the canonical affine cover for flag variety(projective scheme) to prove this result very algebraically | |
| Feb 16, 2010 at 17:30 | history | undeleted |
Pete L. Clark Qiaochu Yuan Anton Geraschenko |
||
| Feb 16, 2010 at 9:19 | history | deleted | Harry Gindi | ||
| Feb 16, 2010 at 9:03 | history | edited | Harry Gindi | CC BY-SA 2.5 |
added 71 characters in body; added 31 characters in body
|
| Feb 16, 2010 at 8:56 | history | asked | Harry Gindi | CC BY-SA 2.5 |