Periodic simple modules for \(SU_ 3(q\) 2) in the describing characteristic p\(\neq 2\)
Considering the special-unitary resp. linear groups \(G=SU_ 3(q^ 2)\) resp. \(H=SL_ 3(p^ n)\) it is easy to see that they have isomorphic Sylow p-subgroups \((q=p^ n)\). Due to Steinberg it is well known, that G and H have exactly one simple projective KG, resp. KH-module, which is called ST, over the algebraic-closed field K with characteristic p. A KG- (KH-)-module M is called periodic, if there is a natural number \(r>1\) with \(\Omega ^ r(M)=\Omega (M)\), where \(\Omega ^ i(M)\) denotes the i- th Heller-module, that is the kernel of the i-th projective cover for a minimal projective resolution of M. I. Janiszczak showed, that for H there are no simple periodic KG-modules except ST, which is projective. It is therefore amazing that, in spite of the similar p-local structure of G and H, there are always periodic non- projective simple modules for G if \(p\neq 2\). The proof requires the theory of module-varieties as developed by Carlson et. al. Also a detailed investigation of the action of a p-Sylow group on irreducible Weyl-modules of G is given. This goes back to the polynomial representations of \(Gl_ m(k)\) as developed by Green and Carter-Lusztig. With a concrete formula for Garnir-relations one can determine the complete matrix-representation of a p-Sylow group of G on Weyl-modules. Finally, the module-varieties of some appropriate simple Weyl-modules can be calculated and, using Steinberg's tensorproduct theorem, simple periodic non-projective KG-modules can be constructed.
scientific article; zbMATH DE number 4021445