On Extensions of Simple Modules over Symmetric and Algebraic Groups

Abstract

Here and below the cohomology of algebraic groups is the rational cohomology (see [13, I.4]). This result is quite useful for calculating some Ext-groups between simple G-modules. For example, it immediately implies that ExtGLλ‘; Lλ‘‘ = 0: We would be interested in having a result similar to (1) for the symmetric group 6n: For a partition λ of n we denote by S the corresponding Specht module, and, if λ is p-regular, we write D for the corresponding irreducible module over F6n (see [11]). It is known that the Specht modules play a role similar to that of Weyl modules for algebraic groups. So we are interested in results connecting Ext6nD;D‘ and Hom6nrad Sλ;Dμ‘. The first main result of the paper is