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| In [[mathematics|mathematical]] field of [[representation theory]], a '''quaternionic representation''' is a [[group representation|representation]] on a [[complex number|complex]] vector space ''V'' with an invariant [[quaternionic structure]], i.e., an [[antilinear]] [[equivariant map]]
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| :<math>j\colon V\to V\,</math> | |
| which satisfies
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| :<math>j^2=-1.\,</math> | | My blog post :: Vi listar alla senaste bästa nya casinon 2014! ([http://wikist.org/Wondering_How_To_Make_Your_Nya_Online_Casino_Rock_Read_This lignende nettside]) |
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| Together with the imaginary unit ''i'' and the antilinear map ''k'' := ''ij'', ''j'' equips ''V'' with the structure of a [[quaternionic vector space]] (i.e., ''V'' becomes a [[module (mathematics)|module]] over the [[division algebra]] of [[quaternion]]s). From this point of view, quaternionic representation of a [[group (mathematics)|group]] ''G'' is a [[group homomorphism]] ''φ'': ''G'' → GL(''V'', '''H'''), the group of invertible quaternion-linear transformations of ''V''. In particular, a quaternionic matrix representation of ''g'' assigns a [[square matrix]] of quaternions ''ρ''(g) to each element ''g'' of ''G'' such that ''ρ''(e) is the identity matrix and
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| :<math>\rho(gh)=\rho(g)\rho(h)\text{ for all }g, h \in G.\,</math> | |
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| Quaternionic representations of [[associative algebra|associative]] and [[Lie algebra]]s can be defined in a similar way.
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| ==Properties and related concepts==
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| If ''V'' is a [[unitary representation]] and the quaternionic structure ''j'' is a unitary operator, then ''V'' admits an invariant complex symplectic form ''ω'', and hence is a [[symplectic representation]]. This always holds if ''V'' is a representation of a [[compact group]] (e.g. a [[finite group]]) and in this case quaternionic representations are also known as symplectic representations. Such representations, amongst [[irreducible representation]]s, can be picked out by the [[Frobenius-Schur indicator]].
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| Quaternionic representations are similar to [[real representation]]s in that they are isomorphic to their [[complex conjugate representation]]. Here a real representation is taken to be a complex representation 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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| A representation which is isomorphic to its complex conjugate, but which is not a real representation, is sometimes called a '''pseudoreal representation'''.
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| Real and pseudoreal representations of a group ''G'' can be understood by viewing them as representations of the real [[group algebra]] '''R'''[''G'']. Such a representation will be a direct sum of central simple '''R'''-algebras, which, by the [[Artin-Wedderburn theorem]], must be matrix algebras over the real numbers or the quaternions. Thus a real or pseudoreal representation is a direct sum of irreducible real representations and irreducible quaternionic representations. It is real if no quaternionic representations occur in the decomposition.
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| == Examples ==
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| A common example involves the quaternionic representation of [[rotation]]s in three dimensions. Each (proper) rotation is represented by a quaternion with [[unit norm]]. There is an obvious one-dimensional quaternionic vector space, namely the space '''H''' of quaternions themselves under left multiplication. By restricting this to the unit quaternions, we obtain a quaternionic representation of the [[spinor group]] Spin(3).
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| This representation ''ρ'': Spin(3) → GL(1,'''H''') also happens to be a unitary quaternionic representation because
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| :<math>\rho(g)^\dagger \rho(g)=\mathbf{1}\,</math>
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| for all ''g'' in Spin(3).
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| Another unitary example is the [[spin representation]] of Spin(5). An example of a nonunitary quaternionic representation would be the two dimensional irreducible representation of Spin(5,1).
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| More generally, the spin representations of Spin(''d'') are quaternionic when ''d'' equals 3 + 8''k'', 4 + 8''k'', and 5 + 8''k'' dimensions, where ''k'' is an integer. In physics, one often encounters the [[spinor]]s of Spin(''d'', 1). These representations have the same type of real or quaternionic structure as the spinors of Spin(''d'' − 1).
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| Among the compact real forms of the simple Lie groups, irreducible quaternionic representations only exist for the Lie groups of type ''A''<sub>4''k''+1</sub>, ''B''<sub>4''k''+1</sub>, ''B''<sub>4''k''+2</sub>, ''C''<sub>''k''</sub>, ''D''<sub>4''k''+2</sub>, and ''E''<sub>7</sub>.
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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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| ==See also==
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| * [[Symplectic vector space]]
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| [[Category:Representation theory]]
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