In the mathematical field of representation theory a real representation is usually a representation on a real vector space U, but it can also mean a representation on a complex vector space V with an invariant real structure, i.e., an antilinear equivariant map which satisfies In physics, where representations are often viewed concretely in terms of matrices, a real representation is one in which the entries of the matrices representing the group elements are real numbers. These matrices can act either on real or complex column vectors. which satisfies
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| - Reelle Darstellung (de)
- Real representation (en)
- Representação real (pt)
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| - In der Mathematik sind reelle Darstellungen ein Begriff der Darstellungstheorie mit zahlreichen Anwendungen in Physik und Mathematik. Er bezeichnet Darstellungen auf einem komplexen Vektorraum, die durch Tensorieren mit den komplexen Zahlen aus einer Darstellung auf einem reellen Vektorraum entstanden sind. (de)
- No matemático campo da teoria da representação uma representação real é normalmente uma representação sobre um espaço vetorial real U, mas também pode significar uma representação sobre um espaço vetorial complexo V com uma invariante, i.e., um o qual satisfaz (pt)
- In the mathematical field of representation theory a real representation is usually a representation on a real vector space U, but it can also mean a representation on a complex vector space V with an invariant real structure, i.e., an antilinear equivariant map which satisfies In physics, where representations are often viewed concretely in terms of matrices, a real representation is one in which the entries of the matrices representing the group elements are real numbers. These matrices can act either on real or complex column vectors. which satisfies (en)
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| - In der Mathematik sind reelle Darstellungen ein Begriff der Darstellungstheorie mit zahlreichen Anwendungen in Physik und Mathematik. Er bezeichnet Darstellungen auf einem komplexen Vektorraum, die durch Tensorieren mit den komplexen Zahlen aus einer Darstellung auf einem reellen Vektorraum entstanden sind. (de)
- In the mathematical field of representation theory a real representation is usually a representation on a real vector space U, but it can also mean a representation on a complex vector space V with an invariant real structure, i.e., an antilinear equivariant map which satisfies The two viewpoints are equivalent because if U is a real vector space acted on by a group G (say), then V = U⊗C is a representation on a complex vector space with an antilinear equivariant map given by complex conjugation. Conversely, if V is such a complex representation, then U can be recovered as the fixed point set of j (the eigenspace with eigenvalue 1). In physics, where representations are often viewed concretely in terms of matrices, a real representation is one in which the entries of the matrices representing the group elements are real numbers. These matrices can act either on real or complex column vectors. A real representation on a complex vector space is isomorphic to its complex conjugate representation, but the converse is not true: a representation which is isomorphic to its complex conjugate but which is not real is called a pseudoreal representation. An irreducible pseudoreal representation V is necessarily a quaternionic representation: it admits an invariant quaternionic structure, i.e., an antilinear equivariant map which satisfies A direct sum of real and quaternionic representations is neither real nor quaternionic in general. A representation on a complex vector space can also be isomorphic to the dual representation of its complex conjugate. This happens precisely when the representation admits a nondegenerate invariant sesquilinear form, e.g. a hermitian form. Such representations are sometimes said to be complex or (pseudo-)hermitian. (en)
- No matemático campo da teoria da representação uma representação real é normalmente uma representação sobre um espaço vetorial real U, mas também pode significar uma representação sobre um espaço vetorial complexo V com uma invariante, i.e., um o qual satisfaz (pt)
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