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Is there any software that will compute cohomology of vector bundles (or just line bundles) on flag manifolds?

The only one I know of is Macaulay2, via the Schubert2 package, but it works with what it calls "abstract varieties", which are really just the intersection rings over $\mathbb{Q}$, so it's explicitly limited to characteristic zero. I'm interested in (among other things) bad behavior at small prime characteristics.

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In prime characteristic, what is known theoretically gives at least the broad outlines of a complicated picture in which cohomology of some line bundles on flag varieties is nonzero in more than one degree. I don't know of any relevant software, but it's risky to try computing this stuff without some history and theory at hand. H.H. Andersen's work starting in the late 1970s has the best results, which suggest strong connections with Kazhdan-Lusztig polynomials for an affine Weyl group. Some examples for small $p$ exist but are hard to interpret in isolation. – Jim Humphreys May 30 '10 at 20:39
Thanks, Jim. Are the examples you're thinking of in Andersen's papers, or more widely scattered? – Graham Leuschke May 30 '10 at 23:55
Can you clarify what you mean by "compute"? The difficulty, as far as I know, is that as a $G$-module, cohomology of a line bundle has complicated structure. But perhaps less would suffice for your purposes. – Victor Protsak May 31 '10 at 4:29
More details are given in a formal answer. By the way, a tag like algebraic-groups might be helpful for your question, since so many techniques for studying vector bundles on flag varieties rely on algebraic groups as well as algebraic geometry. – Jim Humphreys May 31 '10 at 13:12
Victor, for an easy example of what I'd like to compute, take to be the tautological bundle (sub- or quotient, your choice) on $\mathrm{Gr}(2,4)$, and work in char 2 . What are the sheaf cohomologies of, say, $\bigwedge^2\mathcal{Q}^* \otimes \mathrm{Sym}_2\mathcal{Q}$? (In fact, for this particular example I can show that there is no non-zero cohomology, but there are lots of similar examples built out of standard multilinear operations that I can't see yet.) – Graham Leuschke Jun 1 '10 at 0:34
up vote 5 down vote accepted

To answer the original question explicitly, there seems to be no relevant software in prime characteristic. Nor is there any on the horizon, unless the theory developed so far becomes much more definitive. In the setting of flag varieties, general principles show that Euler characters in characteristic $p$ are the same as in characteristic 0 for line bundles (etc.) because the objects involved have compatible $\mathbb{Z}$-forms. Kempf even showed that for a dominant line bundle, sheaf cohomology vanishes except for degree 0; so the formal character and dimension of the global section module are given by Weyl's formula. Similarly for the Serre dual, but there are some systematic patterns (based on alcoves for an affine Weyl group relative to $p$) in which some other line bundles have nonvanishing cohomology in multiple degrees. This appears only for weights "close to" Weyl chamber walls and is conjecturally due to failure of "cancellation" in Jantzen-Andersen filtrations near walls.

A moral of the existing work on line bundles and some other vector bundles is that module structure seems needed to understand the vanishing behavior of cohomology. For small $p$ one lacks analogues of Lusztig's conjectures on characters of the simple modules, which may be an added obstacle. Even small calculations are very difficult, for example for small primes in rank 2. Some have occurred in the literature, but there is no mechanical method to generate them.

Most work was done in the 1980s, following a thesis by Mumford's student W.L. Griffith Jr. showing a few counterexamples to the Borel-Weil-Bott picture for some flag varieties and small $p$, as well as Henning Andersen's MIT thesis around the same time. Full references before 1990 appear in my short conference survey: MR1131312 (92k:20084) 20G10 (14M17 20G05) Humphreys, J. E. (1-MA), Cohomology of line bundles on flag varieties in prime characteristic. Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), 193–204, Manoj Prakashan, Madras, 1991.

A couple of recent papers by Steve Donkin focus mostly on SL(3):

MR1958906 (2004f:20083) 20G05 (14M15 20G10) Donkin, Stephen (4-LNDQM) Anote on the characters of the cohomology of induced vector bundles on G/B in characteristic p. J. Algebra 258 (2002), no. 1, 255–274.

MR2275364 (2008a:20077) 20G10 (14L30 14M15) Donkin, Stephen (4-YORK), The cohomology of line bundles on the three-dimensional flag variety. J. Algebra 307 (2007), no. 2, 570–613.

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