Real Subpairs and Frobenius-schur Indicators of Characters in 2-blocks

Let B be a real 2-block of a finite group G. A defect couple of B is a certain pair (D, E) of 2-subgroups of G, such that D a defect group of B, and D ≤ E. The block B is principal if E = D; otherwise [E : D] = 2. We show that (D, E) determines which B-subpairs are real. The involution module of G arises from the conjugation action of G on its involutions. We outline how (D, E) influences the vertices of components of the involution module that belong to B. These results allow us to enumerate the Frobenius-Schur indicators of the irreducible characters in B, when B has a dihedral defect group. The answer depends both on the decomposition matrix of B and on a defect couple of B. We also determine the vertices of the components of the involution module of B. Date: November 7, 2008. 1991 Mathematics Subject Classification. 20C20.