The canonical bilinear form on a finite root pairing #
Given a finite root pairing, we define a canonical map from weight space to coweight space, and the corresponding bilinear form. This form is symmetric and Weyl-invariant, and if the base ring is linearly ordered, then the form is root-positive, positive-semidefinite on the weight space, and positive-definite on the span of roots. From these facts, it is easy to show that Coxeter weights in a finite root pairing are bounded above by 4. Thus, the pairings of roots and coroots in a crystallographic root pairing are restricted to a small finite set of possibilities. Another application is to the faithfulness of the Weyl group action on roots, and finiteness of the Weyl group.
Main definitions: #
Polarization
: A distinguished linear map from the weight space to the coweight space.RootForm
: The bilinear form on weight space corresponding toPolarization
.
Main results: #
polarization_self_sum_of_squares
: The inner product of any weight vector is a sum of squares.rootForm_reflection_reflection_apply
:RootForm
is invariant with respect to reflections.rootForm_self_smul_coroot
: The inner product of a root with itself times the corresponding coroot is equal to two times Polarization applied to the root.rootForm_self_non_neg
:RootForm
is positive semidefinite.
References: #
- [N. Bourbaki, Lie groups and Lie algebras. Chapters 4--6][bourbaki1968]
- [M. Demazure, SGA III, Exposé XXI, Données Radicielles][demazure1970]
TODO (possibly in other files) #
- Weyl-invariance
- Faithfulness of Weyl group action, and finiteness of Weyl group, for finite root systems.
- Relation to Coxeter weight. In particular, positivity constraints for finite root pairings mean we restrict to weights between 0 and 4.
An invariant linear map from weight space to coweight space.
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An invariant linear map from coweight space to weight space.
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An invariant inner product on the weight space.
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- P.RootForm = ∑ i : ι, LinearMap.smulRight (P.coroot' i) (P.coroot' i)
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An invariant inner product on the coweight space.
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- P.CorootForm = ∑ i : ι, LinearMap.smulRight (P.root' i) (P.root' i)
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This is SGA3 XXI Lemma 1.2.1 (10), key for proving nondegeneracy and positivity.
SGA3 XXI Prop. 2.3.1