Timeline for Integral cohomology of $G/N(T)$
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 28, 2016 at 8:00 | vote | accept | Alex Fok | ||
| S May 28, 2016 at 7:59 | history | bounty ended | Alex Fok | ||
| S May 28, 2016 at 7:59 | history | notice removed | Alex Fok | ||
| May 24, 2016 at 10:05 | comment | added | David E Speyer | I just realized that it isn't clear to me that this is the map for which stabilization occurs. | |
| May 24, 2016 at 9:44 | comment | added | David E Speyer | Something neat happens when $p=2$. We have a short exact sequence $0 \to \mathbb{Z} \to \mathbb{Z}^n \to H^2(\mathcal{F}) \to 0$, with the obvious actions. We can describe $\mathbb{Z}^n$ as $\mathrm{Ind}_{S_{n-1}}^{S_n} \mathbb{Z}$, so $H^q(S_n, \mathbb{Z}^n) \cong H^q(S_{n-1}, \mathbb{Z})$. Thus, the terms in $p=2$ measure the failure of $H^q(S_{n-1}, \mathbb{Z}) \to H^q(S_n, \mathbb{Z})$ to be be an isomorphism -- in other words, the lack of stabilization. If we fix $q$ and send $n \to \infty$, they are eventually $0$. | |
| May 24, 2016 at 9:34 | comment | added | David E Speyer | Here is a sign that this may be hard. We have a spectral sequence $H^q(W, H^p(\mathcal{F})) \to H^{p+q}(\mathcal{F}/W)$ where the left hand side is group cohomology. Take a look at the $p=0$ row for $SU(n)$: tables of $H^q(S_n, \mathbb{Z})$ can be found at groupprops.subwiki.org/wiki/… . They aren't simple... | |
| May 24, 2016 at 8:12 | answer | added | Neil Strickland | timeline score: 4 | |
| May 23, 2016 at 15:46 | answer | added | Sean Lawton | timeline score: 10 | |
| S May 20, 2016 at 9:32 | history | bounty started | Alex Fok | ||
| S May 20, 2016 at 9:32 | history | notice added | Alex Fok | Authoritative reference needed | |
| May 18, 2016 at 17:31 | comment | added | Bombyx mori | @AlexFok: Not sure if this helps: mathoverflow.net/questions/78717/…. For $SU(n)$, I know what the basis elements of $R(T)$ over $R(G)$ is, but I do not know much about the Lie group integral cohomology. | |
| May 18, 2016 at 17:23 | comment | added | Bombyx mori | @AlexFok: I vaguely remember this is one of the questions he proposed in 1970s or 1980s. In my undergraduate thesis defense I listed it and my advisor claimed it was long resolved (thus I was mistaken that it was open). Unfortunately he did not give a reference. | |
| May 18, 2016 at 17:13 | comment | added | Alex Fok | @Bombyx mori Could you please give me a reference? | |
| May 18, 2016 at 17:07 | comment | added | Bombyx mori | If I am not mistaken this is worked out by Manin already? | |
| May 18, 2016 at 16:45 | history | edited | Alex Fok | CC BY-SA 3.0 |
added 9 characters in body
|
| May 18, 2016 at 16:11 | history | edited | Alex Fok | CC BY-SA 3.0 |
added 3 characters in body; edited title
|
| May 18, 2016 at 16:11 | comment | added | Alex Fok | I agree. I changed the notation for the normalizer accordingly. | |
| May 18, 2016 at 13:14 | comment | added | Allen Knutson | I advocate using $N(T)$ to denote this group, since $N$ often gets used for the unipotent group. (Admittedly noone would ask about the cohomology of $G/N$ with that meaning, since that $N$ is contractible.) | |
| May 18, 2016 at 8:43 | history | asked | Alex Fok | CC BY-SA 3.0 |