Editing Quaternionic representation (section)

==Properties and related concepts==

If ''V'' is a [[unitary representation]] and the quaternionic structure ''j'' is a unitary operator, then ''V'' admits an invariant complex symplectic form ''&omega;'', and hence is a [[symplectic representation]]. This always holds if ''V'' is a representation of a [[compact group]] (e.g. a [[finite group]]) and in this case quaternionic representations are also known as symplectic representations. Such representations, amongst [[irreducible representation]]s, can be picked out by the [[Frobenius-Schur indicator]].

Quaternionic representations are similar to [[real representation]]s in that they are isomorphic to their [[complex conjugate representation]]. Here a real representation is taken to be a [[complex representation]] with an invariant [[real structure]], i.e., an [[antilinear]] [[equivariant map]]

:<math>j\colon V\to V</math>

which satisfies

:<math>j^2=+1.</math>

A representation which is isomorphic to its complex conjugate, but which is not a real representation, is sometimes called a '''pseudoreal representation'''.

Real and pseudoreal representations of a group ''G'' can be understood by viewing them as representations of the real [[group ring|group algebra]] '''R'''[''G'']. Such a representation will be a direct sum of central simple '''R'''-algebras, which, by the [[Artin-Wedderburn theorem]], must be matrix algebras over the real numbers or the quaternions. Thus a real or pseudoreal representation is a direct sum of irreducible real representations and irreducible quaternionic representations. It is real if no quaternionic representations occur in the decomposition.