NONREALIZABILITY OF CERTAIN REPRESENTATIONS IN FUSION SYSTEMS

For a finite abelian p-group A and a subgroup G = Aut(A), we say that the pair (G, A) is fusion realizable if there is a saturated fusion system F over a finite p-group S = A such that CS(A) = A, AutF (A) = G as subgroups of Aut(A), and A not greater than F. In this paper, we develop tools to show that certain representations are not fusion realizable in this sense. For example, we show, for p = 2 or 3 and G one of the Mathieu groups, that the only FpG-modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases their duals.