Editing Real representation

{{Short description|Type of representation in representation theory}}
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]] [[Representation_theory#Equivariant_maps_and_isomorphisms|equivariant map]]

:<math>j\colon V\to V</math>
which satisfies
:<math>j^2=+1.</math>
The two viewpoints are equivalent because if ''U'' is a real vector space acted on by a group ''G'' (say), then ''V'' = ''U''&otimes;'''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
:<math>j\colon V\to V</math>
which satisfies
:<math>j^2=-1.</math>
A [[direct sum of representations|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.

==Frobenius-Schur indicator==
{{Main|Frobenius-Schur indicator}}

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
:<math>\int_{g\in G}\chi(g^2)\,d\mu</math>
where ''&chi;'' is the character of the representation and ''μ'' is the [[Haar measure]] with μ(''G'') = 1. For a finite group, this is given by
:<math>{1\over |G|}\sum_{g\in G}\chi(g^2).</math>
The indicator may take the values 1, 0 or &minus;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 &minus;1, the representation is quaternionic.

==Examples==

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]].

All representations of the [[special orthogonal group|rotation group]]s on odd-dimensional spaces are real, since they all appear as subrepresentations of [[tensor products]] of copies of the fundamental representation, which is real.

Further examples of real representations are the [[spinor]] representations of the [[spin group]]s in 8''k''&minus;1, 8''k'', and 8''k''+1 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]] and [[Bott periodicity]].

==Notes==
{{reflist|1}}

==References==
*{{Fulton-Harris}}.
*{{citation | first=Jean-Pierre | last=Serre | title=Linear Representations of Finite Groups | publisher=Springer-Verlag | year=1977 | isbn=978-0-387-90190-9 | url-access=registration | url=https://archive.org/details/linearrepresenta1977serr }}.

[[Category:Representation theory]]