| Pseudoreal Representation |
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Information AboutPseudoreal Representation |
| CATEGORIES ABOUT SYMPLECTIC REPRESENTATION | |
| representation theory of groups | |
| symplectic geometry | |
| theoretical physics | |
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When it comes to representations that are not irreducible, one could give an example of a Direct Sum of a real representation and a symplectic representation, as satisfying the stated condition to be ''pseudoreal''; this however is less useful. An irreducible group representation that is neither real nor symplectic is a Complex Representation . Approaching the question from the direction of Schur's Lemma , i.e. Module Endomorphism s for the representation space, there exists for a symplectic representation an Antilinear Map : that commutes with the elements of the Group , but it satisfies :. Pseudoreal representations are often called quaternionic representations because the group elements can be expressed as matrices whose entries are quaternions. Another way to look at this is to look at the real Group Algebra , and identify it as a direct sum of simple ''R''-algebras. These will be ( Artin-Wedderburn Theorem ) matrix algebras over the real numbers ''or'' the quaternions. The latter case is responsible for the phenomenon of pseudoreal representation. Examples of pseudoreal representations are the Spinor s in , and dimensions where ''k'' is an integer. SEE ALSO |
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