|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| 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]]
| | Some person who wrote some article is called Leland but it's not some of the most masucline name on the internet. To go to karaoke is the thing david loves most of . He works as a cashier. His wife and him live all through Massachusetts and he comes with everything that he [http://www.adobe.com/cfusion/search/index.cfm?term=&specifications&loc=en_us&siteSection=home specifications] there. He's not godd at design but locate want to check that website: http://prometeu.net<br><br>Also visit my blog; [http://prometeu.net clash of clans hack tool] |
| :<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''⊗'''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== | |
| | |
| 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 ''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
| |
| :<math>{1\over |G|}\sum_{g\in G}\chi(g^2).</math> | |
| 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.
| |
| | |
| ==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 are real, since they all appear as subrepresentations of [[tensor product]]s 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''−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]].
| |
| | |
| ==Notes==
| |
| {{reflist|1}}
| |
| | |
| ==References==
| |
| *{{Fulton-Harris}}.
| |
| *{{citation |first=Jean-Pierre|last= Serre | title=Linear Representations of Finite Groups | publisher=Springer-Verlag | year=1977 | isbn= 0-387-90190-6}}.
| |
| | |
| [[Category:Representation theory]]
| |
Some person who wrote some article is called Leland but it's not some of the most masucline name on the internet. To go to karaoke is the thing david loves most of . He works as a cashier. His wife and him live all through Massachusetts and he comes with everything that he specifications there. He's not godd at design but locate want to check that website: http://prometeu.net
Also visit my blog; clash of clans hack tool