ABSTRACT
Let SL2 be an algebraic group defined over an algebraically closed field k of characteristic p > 0. In this paper, we provide a closed formula for for Weyl SL2-modules V(m) when n ≤ 2p − 3. For n > 2p − 3, an exponential bound, only depending on n, is obtained for
. Analogous results are also established for the extension spaces
between Weyl modules V(m1) and V(m2). As a by-product, our results and techniques give explicit upper bounds for the dimensions of cohomology of the Specht modules of symmetric groups, and the cohomology of simple modules of SL2 and the finite group of Lie type
.
Acknowledgments
The second author would like to thank Chris Bendel, Dan Nakano, Jon Carlson, and Vanessa Miemietz for useful discussions. We are also grateful to the anonymous referee for his/her comments/suggestions improving the manuscript.
Notes
1We are aware of some technical errors in the paper. From our communication with Jon Carlson, the errors, which are only about the second cohomology computation, do not affect the result we are using here.