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| In mathematics, the '''root datum''' ('''donnée radicielle''' in French) of a connected split [[reductive group|reductive]] [[algebraic group]] over a field is a generalization of a [[root system]] that determines the group up to isomorphism. They were introduced by [[Michel Demazure]] in [[Grothendieck's Séminaire de géométrie algébrique|SGA III]], published in 1970.
| | The writer is called Araceli Gulledge. Delaware is the only location I've been residing in. To play badminton is something he really enjoys doing. Managing people is what I do in my day job.<br><br>Feel free to visit my website ... [http://newdayz.de/index.php?mod=users&action=view&id=16038 newdayz.de] |
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| ==Definition==
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| A '''root datum''' consists of a quadruple
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| :<math>(X^\ast, \Phi, X_\ast, \Phi^\vee)</math>,
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| where
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| * <math>X^\ast</math> and <math>X_\ast</math> are free abelian groups of finite [[Rank (linear algebra)|rank]] together with a [[perfect pairing]] between them with values in <math>\mathbb{Z}</math> which we denote by ( , ) (in other words, each is identified with the [[dual lattice]] of the other).
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| * <math>\Phi</math> is a finite subset of <math>X^\ast</math> and <math>\Phi^\vee</math> is a finite subset of <math>X_\ast</math> and there is a bijection from <math>\Phi</math> onto <math>\Phi^\vee</math>, denoted by <math>\alpha\mapsto\alpha^\vee</math>.
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| * For each <math>\alpha</math>, <math>(\alpha, \alpha^\vee)=2</math>.
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| * For each <math>\alpha</math>, the map <math>x\mapsto x-(x,\alpha^\vee)\alpha</math> induces an automorphism of the root datum (in other words it maps <math>\Phi</math> to <math>\Phi</math> and the induced action on <math>X_\ast</math> maps <math>\Phi^\vee</math> to <math>\Phi^\vee</math>)
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| The elements of <math>\Phi</math> are called the '''roots''' of the root datum, and the elements of <math>\Phi^\vee</math> are called the '''coroots'''. The elements of <math>X^\ast</math> are sometimes called '''[[Weight_(representation_theory)|weights]]''' and those of <math>X_\ast</math> accordingly '''coweights'''.
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| If <math>\Phi</math> does not contain <math>2\alpha</math> for any <math>\alpha\in\Phi</math>, then the root datum is called '''reduced'''.
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| ==The root datum of an algebraic group==
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| If ''G'' is a reductive algebraic group over an [[algebraically closed field]] ''K'' with a split maximal torus ''T'' then its '''root datum''' is a quadruple
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| :(''X''<sup>*</sup>, Φ, ''X''<sub>*</sub>, Φ<sup>v</sup>),
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| where
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| *''X''<sup>*</sup> is the lattice of characters of the maximal torus,
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| *''X''<sub>*</sub> is the dual lattice (given by the 1-parameter subgroups),
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| *Φ is a set of roots,
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| *Φ<sup>v</sup> is the corresponding set of coroots.
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| A connected split reductive algebraic group over ''K'' is uniquely determined (up to isomorphism) by its root datum, which is always reduced. Conversely for any root datum there is a reductive algebraic group. A root datum contains slightly more information than the [[Dynkin diagram]], because it also determines the center of the group.
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| For any root datum (''X''<sup>*</sup>, Φ,''X''<sub>*</sub>, Φ<sup>v</sup>), we can define a '''dual root datum''' (''X''<sub>*</sub>, Φ<sup>v</sup>,''X''<sup>*</sup>, Φ) by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.
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| If ''G'' is a connected reductive algebraic group over the algebraically closed field ''K'', then its [[Langlands dual group]] <sup>''L''</sup>''G'' is the complex connected reductive group whose root datum is dual to that of ''G''.
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| ==References==
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| *[[Michel Demazure]], Exp. XXI in [http://modular.fas.harvard.edu/sga/sga/3-3/index.html SGA 3 vol 3]
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| *[[T. A. Springer]], [http://www.ams.org/online_bks/pspum331/pspum331-ptI-1.pdf ''Reductive groups''], in [http://www.ams.org/online_bks/pspum331/ ''Automorphic forms, representations, and L-functions'' vol 1] ISBN 0-8218-3347-2
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| [[Category:Representation theory]]
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| [[Category:Algebraic groups]]
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The writer is called Araceli Gulledge. Delaware is the only location I've been residing in. To play badminton is something he really enjoys doing. Managing people is what I do in my day job.
Feel free to visit my website ... newdayz.de