Quaternionic representation

In mathematics and theoretical physics, a pseudoreal representation is a group representation that is equivalent to its complex conjugate, but that is not a real representation.

In other words, there exists an antilinear map $j:V\to V$ that commutes with the elements of the group, but it satisfies $j^{2}=-1$. Pseudoreal representations are often called quaternionic representations because the group elements can be expressed as matrices whose entries are quaternions.

Examples of pseudoreal representations are the spinors in $3+8k,4+8k,5+8k$ dimensions where $k\in Z$.

A group representation that is neither real nor pseudoreal is called a complex representation.