Timeline for Presentation of the cohomology of generalized flag varieties as graded ranks of rings of symmetric polynomials
Current License: CC BY-SA 2.5
9 events
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| Oct 2, 2010 at 9:05 | comment | added | Hanno | Ok, thanks alot for your help, Ben! | |
| Oct 2, 2010 at 8:07 | comment | added | Ben Webster♦ | well, the dimension is right by Schubert decomposition, so it's enough to show injectivity or surjectivity. For surjectivity, there are a couple of arguments; you can use Eilenberg-Moore as Torsten does, or I think you could use Kirwan surjectivity. Localization in equivariant cohomology would also work. It's like that there's some more down to earth method I'm just not thinking of as well, like writing down a explicit set of Chern classes which are upper-triangular with respect to Schubert classes, but I couldn't point you to such an argument. | |
| Oct 2, 2010 at 7:30 | vote | accept | Hanno | ||
| Oct 2, 2010 at 7:17 | comment | added | Hanno |
Now I figured out how to compute the cohomology of $F(m_1,...,m_k)$ wit the knowledge of $F(m_1+...+m_k)$ using Leray Hirsch for the fibre bundle $F(m_1)\times ...\times F(m_k)\to F(m_1+...+m_k)\to F(m_1,...,m_k)$. Concerning Chern roots: So you want to consider the Chern classes as the elementary symmetric polynomials in the Chern roots computed after choice of an arbitrary map splitting the bundle after pullback? That's nice! But how would you proceed to show that there are no more relations, and that the chern classes generate the cohomology?
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| Oct 1, 2010 at 21:25 | comment | added | Ben Webster♦ | Roughly, yes, though you can state everything more canonically in terms of Chern roots. | |
| Oct 1, 2010 at 21:20 | history | edited | Ben Webster♦ | CC BY-SA 2.5 |
added 749 characters in body; deleted 749 characters in body
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| Oct 1, 2010 at 19:44 | comment | added | Hanno |
First, what do you mean with "The Chern classes of these bundles are the elementary symmetric polynomials in the variables corresponding to $S_{m_i}$"? Do you consider the cohomology of $F(m_1,...,m_k)$ as a subring of the cohomology of the full flag variety F via the canonical map $F\to F(m1,...,mk)$? Then what you say is a consequence of the Whitney sum formula, too, isn't it?
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| Oct 1, 2010 at 19:34 | comment | added | Hanno |
@Ben: Thanks alot, this is awesome. I didn't notice that for a homomorphism of positively graded connected k-algebras $R_{\bullet}\to S_{\bullet}$ making $S_{\bullet}$ into a free graded $R_{\bullet}$-module of finite rank, the q-rank equals the q-dimension of the quotient S_{\bullet}/⟨R+⟩. That's nice :-) Concerning the actual computation of $H^{\ast}(F(m_1,...,m_k);{\mathbb C})$`, I have some questions, see next comment.
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| Oct 1, 2010 at 18:47 | history | answered | Ben Webster♦ | CC BY-SA 2.5 |