Cohomology of SL₂ and related structures

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 $dim{H}^n(S{L}_2,V(m\left)\right)$ for Weyl SL₂-modules V(m) when n ≤ 2p − 3. For n > 2p − 3, an exponential bound, only depending on n, is obtained for $dim{H}^n(S{L}_2,V(m\left)\right)$. Analogous results are also established for the extension spaces ${Ext}_{S{L}_2}^n\left(V\right(m_2),V(m_1\left)\right)$ 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 $S{L}_2\left(p^s\right)$.

KEYWORDS:

• Algebraic groups
• cohomology
• Frobenius kernels
• symmetric groups
• Weyl module

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 20G10
• 20C30
• 20C33

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.