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| In the [[mathematics|mathematical]] field of [[representation theory]] a '''real representation''' is usually a [[group representation|representation]] on a [[real number|real]] [[vector space]] ''U'', but it can also mean a representation on a [[complex number|complex]] vector space ''V'' with an invariant [[real structure]], i.e., an [[antilinear]] [[equivariant map]]
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| :<math>j\colon V\to V\,</math>
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| which satisfies
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| :<math>j^2=+1.\,</math>
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| 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).
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| 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.
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| 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
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| :<math>j\colon V\to V\,</math>
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| which satisfies
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| :<math>j^2=-1.\,</math>
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| A [[direct sum of representations|direct sum]] of real and quaternionic representations is neither real nor quaternionic in general.
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| 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.
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| ==Frobenius-Schur indicator==
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| A criterion (for [[compact group]]s ''G'') for reality of irreducible representations in terms of [[character theory]] is based on the '''[[Frobenius-Schur indicator]]''' defined by
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| :<math>\int_{g\in G}\chi(g^2)\,d\mu</math> | |
| where ''χ'' is the character of the representation and ''μ'' is the [[Haar measure]] with μ(''G'') = 1. For a finite group, this is given by
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| :<math>{1\over |G|}\sum_{g\in G}\chi(g^2).</math>
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| The indicator may take the values 1, 0 or −1. If the indicator is 1, then the representation is real. If the indicator is zero, the representation is complex (hermitian),<ref>Any complex representation ''V'' of a compact group has an invariant hermitian form, so the significance of zero indicator is that there is no invariant nondegenerate complex bilinear form on ''V''.</ref> and if the indicator is −1, the representation is quaternionic.
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| ==Examples==
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| All representation of the [[symmetric group]]s are real (and in fact rational), since we can build a complete set of [[irreducible representations]] using [[Young tableaux]].
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| All representations of the [[special orthogonal group|rotation group]]s are real, since they all appear as subrepresentations of [[tensor product]]s of copies of the fundamental representation, which is real.
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| Further examples of real representations are the [[spinor]] representations of the [[spin group]]s in 8''k''−1, 8''k'', and 1 + 8''k'' dimensions for ''k'' = 1, 2, 3 ... . This periodicity ''[[Modular arithmetic|modulo]]'' 8 is known in mathematics not only in the theory of [[Clifford algebra]]s, but also in [[algebraic topology]], in [[KO-theory]]; see [[spin representation]].
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| ==Notes==
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| {{reflist|1}}
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| ==References==
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| *{{Fulton-Harris}}.
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| *{{citation |first=Jean-Pierre|last= Serre | title=Linear Representations of Finite Groups | publisher=Springer-Verlag | year=1977 | isbn= 0-387-90190-6}}.
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| [[Category:Representation theory]]
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