Root datum: Difference between revisions

Revision as of 18:17, 2 July 2013

Revision as of 12:34, 5 May 2014

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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>)

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_Weight (representation_theoryrepresentation theory)|weights]]''' and those of <math>X_\ast</math> accordingly '''coweights'''.

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

*&Phi; is a set of roots,

*&Phi;^v is the corresponding set of coroots.

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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==References==

*[[Michel 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]]