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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 (''X'', Ψ, ''X''^∨, Ψ^∨), where:

:<math>(X^\ast, \Phi, X_\ast, \Phi^\vee)</math>,

where

*''X'', ''<math>X''^\ast<sup/math>&or; and <math>X_\ast</supmath> 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'''.

*''X'', ''X''^&or; 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 [[algebraically closed field]] ''<math>K''</math> with a split maximal torus ''<math>T''</math> then its '''root datum''' is a quadruple :(''X''^*, &Delta;,''X''_*, &Delta;^&or;), where

:<math>(X^*, \Phi, X_*, \Phi^{\vee})</math>,

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),

*&Delta;<math>\Phi</math> is a set of roots,

*&Delta;<supmath>&or;\Phi^{\vee}</supmath> is the corresponding set of coroots.

A connected split reductive algebraic group over <math>K</math> 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.

If ''G'' is a reductive algebraic group over a field ''K'' with a split maximal torus ''T'' then its '''root datum''' is a quadruple :(''X''^*, &Delta;,''X''_*, &Delta;^&or;), where

For any root datum <math>(''X''^{^*}, &Delta;\Phi,''X''_{X_*}, &Delta;<sup>&or;\Phi^{\vee})</supmath>), we can define a '''dual root datum''' (''X''<submath>(X_*</sub>, &Delta;^v\Phi^{\vee},''X''^{^*}, &Delta;\Phi)</math> by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.

*''X''^* is the lattice of characters of the maximal torus,

*''X''_* is the dual lattice (given by the 1-parameter subgroups),

*&Delta; is a set of roots,

*&Delta;^&or; is the corresponding set of coroots.

AIf <math>G</math> is a connected reductive algebraic group over anthe algebraically closed field is<math>K</math>, uniquelythen determinedits (up[[Langlands todual isomorphism)group]] by its root datum,<math>{}^L whichG</math> is alwaysthe reduced.complex Conversely for any root datum there is aconnected reductive algebraic group. Awhose root datum containsis slightlydual moreto information than the [[Dynkin diagram]], because it also determines the centerthat of the group<math>G</math>.

==References==

For any root datum (''X''^*, &Delta;,''X''_*, &Delta;^&or;), we can define a '''dual root datum''' (''X''_*, &Delta;^v,''X''^*, &Delta;) by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.

*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 }}

If ''G'' is a connected reductive algebraic group over the algebraically closed field ''K'', then its [[Langlands dual group]] ^''L''''G'' is the complex connected reductive group whose root datum is dual to that of ''G''.

==References==

*M. Demazure, Exp. XXI in [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 0-8218-3347-2

[[Category:Representation theory]]

[[Category:Algebraic groups]]