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I am interested in calculating cohomology of line bundles on flag varieties $G/B$ in positive characteristic. But I really just have a bunch of scattered examples. Does there exist some kind of software that will calculate this for me? For the most part I don't care about the representation structure of the cohomology modules, I just want to know dimensions. Also, I know there are various results on when this cohomology is just like the char. 0 situation, but they won't always apply to my examples. So I don't need general results, just an algorithm.

I just know how to do one example using Macaulay2: irreducible homogeneous bundles on projective space (which are pushforwards of aforementioned line bundles). But I am also interested in things like type D, and homogeneous bundles on Grassmannians.

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Adding a tag algebraic-groups would be useful here. –  Jim Humphreys Oct 14 '11 at 20:22

2 Answers 2

This is a sort of negative-leaning answer to the question about existence of software for your purpose. There is quite a bit of history to the problem in prime characteristic, going back to isolated examples found in the 1970s by Mumford and his 1975 Ph.D. student W.L. Griffith Jr. (Cohomology of Flag Varieties in Characteristic p) which showed that the classical ideas could break down. The rank 2 example $G_2$ lends itself to picture drawing and has been looked at in considerable detail. See the recent updated preprint by Andersen and Kaneda here.

Andersen's clever sheaf cohomology techniques (exploiting the Frobenius map) combined with my more speculative predictions tend to imply that the results depend heavily on Kazhdan-Lusztig theory for the affine Weyl group (of Langlands dual type). Moreover, the non-vanishing of cohomology seems to involve the actual module structure, so dimensions appear only as a byproduct of the study of generic module filtrations crossing Weyl chamber walls. The algebraic group of type $G_2$ already indicates how systematic but complicated the results will be in general, so any computational approach must take this case into account. (The results for $A_2$ and $B_2$ worked out by Andersen following his 1977 MIT thesis On Schubert Varieties in G/B and Bott's Theorem are also subtle, but con't compete with the complexity of $G_2$ whose alcove geometry is richer.)

ADDED: The problem arose in the setting of algebraic geometry, as seen in the thesis work mentioned above. Seshadri wrote up his own version of the $SL_3$ case treated by Larry Griffith, in a typescript Cohomology of line bundles on $SL_3/B$ (Tata Institute, September 28, 1976). I learned about the problem from him the following spring at IAS and formulated my own tentative interpretation in a conference paper that summer. Andersen recovered Griffith's results in a general setting in his 1979 Inventiones paper here. In particular, an extra non-vanishing $H^1$ has a unique simple submodule of specified highest weight. But pinning down the dimension or formal character of this module takes more work, done first by Jantzen (before Kazhdan-Lusztig theory). There may be shortcuts in small cases, but a general algorithmic approach to the flag variety of $SL_3$ gets complicated.

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I sort of suspected that what I want does not exist. But your answer is still great, and I think I can learn a lot from these references. Also, presumably your reference to Mumford's student is to the paper W. Griffith, "Cohomology of flag varieties in characteristic $p$". This gives the complete answer for $A_2$ (at least for when Bott's theorem fails) with a simple rule and I really like it. –  Steven Sam Oct 14 '11 at 21:39
I don't know a lot about this story, but the cohomology of a homogeneous line bundle on a Grassmannian should be equal to the cohomology of a line bundle on the flag variety in type A corresponding to a weight that vanishes on all but one coroot, which is a very special case. Perhaps there is an algorithm in that case? (There might be some helpful comments in Jantzen's Representations of Algebraic Groups as well). –  Chuck Hague Oct 17 '11 at 15:46

This may be in the scope of http://sagemath.org/ .

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