Real Representations of $C_2$-Graded Groups: The Antilinear Theory

We use the structure of finite-dimensional graded algebras to develop the theory of antilinear representations of finite C2-graded groups. A finite C2-graded group is a finite group with a subgroup of index 2. In this theory the subgroup acts linearly, while the other coset acts antilinearly. We introduce antilinear blocks, whose structure is a crucial component of the theory. Among other things, we study characters and Frobenius-Schur indicators. As an example, we describe the antilinear representations of the C2-graded group An≤Sn.