Invariant lattices and modular decomposition of irreducible representations

Abstract

Let p be a prime. Let H be a finite group, and let c be an (absolutely) irreducible character of H. Any thorough study of the p-modular decom- position of c requires passage to some integral representation. If S is a (finite) group acting on H and leaving [ invariant, such an integral represen tation will not be S-invariant in general. It is important to have invariant integral representations for purposes of Clifford theory. In fact, modular decomposition often gives new insights in such “classical” questions like extendability of characters. For this reason we consider the more general situation that H is a normal subgroup of some group G, and let S = G/H. (S still acts on the characters of H and the isomorphism types of H-modules.) We also fix a p-modular system (K, R, k), where