Real subpairs and Frobenius–Schur indicators of characters in 2-blocks

AbstractLet 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 is 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