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In mathematicsmathematical [[group theory]], the root datum ('''donnéeroot radicielledatum''' 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 M.[[Michel Demazure]] in [[Grothendieck's Séminaire de géométrie algébrique|SGA III]], published in 1970.
==Definition==
A '''root datum''' consists of a quadruple
:<math>(X^\ast, \PsiPhi,X^{ X_\vee}ast, \PsiPhi^{\vee}) </math>,
where
* <math> X, X^ {\ vee}ast</math> and <math>X_\ast</math> are free abelian groups of finite [[ Rank (linear algebra)|rank]] together with a [[perfect pairing ]] <math>\langlebetween ,them \ranglewith :values Xin <math>\ times X^{\vee} \rightarrow \mathbfmathbb{Z}</math> betweenwhich themwe denote by ( , ) (in other words, each is identified with the [[dual lattice]] of the other). ▼* <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>.
* For each <math>\alpha</math>, we have: <math> \langle (\alpha, \alpha^ {\vee }\rangle )=2 </math> .▼* For each <math>\alpha</math>, the map <math>x \mapsto x - \langle (x,\alpha^ {\vee } \rangle )\alpha </math> induces an automorphism of the root datum (in other words it maps <math>\ PsiPhi</math> to <math>\ PsiPhi</math> and the induced action on <math> X^{X_\ vee} ast</math> maps <math> \ PsiPhi^ {\vee } </math> to itself.<math>\Phi^\vee</math>)▼
The elements of <math>\ PsiPhi</math> are called the '''roots''' of the root datum, and the elements of <math> \ PsiPhi^ {\vee } </math> are called the '''coroots'''. ▼▲*<math>X, X^{\vee}</math> are free abelian groups of finite [[rank]] together with a perfect pairing <math>\langle , \rangle : X \times X^{\vee} \rightarrow \mathbf{Z}</math> between them (in other words, each is identified with the [[dual lattice]] of the other).
*If <math>\Psi Phi</math> is a finitedoes subsetnot of <math>X</math> andcontain <math>2\Psi^{\vee} alpha</math> isfor a finite subset ofany <math>X^{\vee}</math> and there is a bijection from <math>\Psi</math> onto <math>alpha\Psi^{in\vee}Phi</math>, denotedthen bythe <math>\alpharoot \mapstodatum \alpha^{\vee}</math>is called '''reduced'''.
▲*For each <math>\alpha</math>, we have: <math> \langle \alpha, \alpha^{\vee}\rangle =2 </math>
▲*For each <math>\alpha</math>, the map <math>x \mapsto x - \langle x,\alpha^{\vee} \rangle \alpha </math> induces an automorphism of the root datum (in other words it maps <math>\Psi</math> to <math>\Psi</math> and the induced action on <math>X^{\vee} </math> maps <math> \Psi^{\vee} </math> to itself.
▲The elements of <math>\Psi</math> are called the '''roots''' of the root datum, and the elements of <math> \Psi^{\vee} </math> are called the '''coroots'''.
If <math> \Psi </math> does not contain <math>2 \alpha</math> for any <math>\alpha</math> in <math> \Psi </math> then the root datum is called '''reduced'''.
==The root datum of an algebraic group==
If ''<math>G''</math> is a reductive algebraic group over aan field[[algebraically closed field]] ''<math>K''</math> with a split maximal torus ''<math>T''</math> then its '''root datum''' is a quadruple
:<math>(X^*, \Phi, X_*, \Phi^{\vee})</math>,
:(''X''<sup>*</sup>, Δ,''X''<sub>*</sub>, Δ<sup>v</sup>),
where
*''X''<supmath>X^*</supmath> is the lattice of characters of the maximal torus,
*''X''<submath>X_*</submath> is the dual lattice (given by the 1-parameter subgroups),
*Δ<math>\Phi</math> is a set of roots,
*Δ<supmath>v\Phi^{\vee}</supmath> is the corresponding set of coroots.
A connected split reductive algebraic group over an algebraically closed<math>K</math> field 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.
For any root datum <math>(''X''<sup>^*</sup>, Δ\Phi,''X''<sub> X_*</sub>, Δ<sup>v\Phi^{\vee})</supmath>), we can define a '''dual root datum''' (''X''<submath>(X_*</sub>, Δ<sup>v</sup>\Phi^{\vee},''X''<sup>^*</sup>, Δ\Phi)</math> by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.
If ''<math>G''</math> is a connected reductive algebraic group over the algebraically closed field ''<math>K''</math>, then its [[Langlands dual group]] <supmath>''{}^L'' G</supmath>''G'' is the complex connected reductive group whose root datum is dual to that of ''<math>G''</math>.
==References==
*M.[[Michel Demazure]], Exp. XXI in [https://web.archive.org/web/20011126072304/http://modular.fas.harvard.edu/sga/sga/3-3/index.html SGA 3 vol 3]
*[[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 {{isbn|0-8218-3347-2 }}
[[Category:Representation theory]]
[[Category:Algebraic groups]]
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