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Weyl chamber
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(Definition)
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Let $E$ be a Euclidean vector space, $R\subset E$ a root system, and $R^+\subset R$ a choice of positive roots. We define the positive Weyl chamber (relative to $R^+$ to be the closed set $$\mathcal{C}=\{u\in E\mid (u,\alpha)\geq 0\text{ for all }\alpha\in R^+\}.$$ A weight which lies inside the positive Weyl chamber is called dominant.
The interior of $\mathcal{C}$ is a fundamental domain for the action of the Weyl group on $E$ The image $w(\mathcal{C})$ of $\mathcal{C}$ under the any element $w$ of the Weyl group is called a Weyl chamber. The Weyl group $W$ acts simply transitively on the set of Weyl chambers.
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"Weyl chamber" is owned by rmilson. [ full author list (3) | owner history (2) ]
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positive Weyl chamber, dominant weight |
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Cross-references: image, Weyl group, action, domain, interior, weight, closed set, positive roots, root system, Euclidean vector space
There is 1 reference to this entry.
This is version 5 of Weyl chamber, born on 2002-12-05, modified 2004-04-08.
Object id is 3661, canonical name is WeylChamber.
Accessed 5848 times total.
Classification:
| AMS MSC: | 17B20 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Simple, semisimple, reductive ) |
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Pending Errata and Addenda
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