In the other case, the invariant bilinear form is skew-symmetric. The same argument produces a -equivariant antilinear map with .
Definition 16.5.1 (Quaternionic structure). A quaternionic structure on a complex vector space is a -antilinear map with . A representation with such a -equivariant is called a quaternionic representation.
Such a makes into a module over the quaternions , the division algebra (noncommutative ring where every nonzero element has a multiplicative inverse) with . Concretely, with ,, so an -module is a -vector space with an antilinear map (“right multiplication by ”) satisfying .
Note. A quaternionic representation has real-valued (since ) but does not come from a real representation. The three cases for an irreducible with real character are:
admits a -invariant nondegenerate symmetric bilinear form is real .
admits a -invariant nondegenerate skew-symmetric bilinear form is quaternionic carries a -equivariant antilinear with .
And if is not real-valued, then and is neither real nor quaternionic.