On convex hulls of orbits of Coxeter groups and Weyl groups
Hofmann G, Neeb KH (2014)
Publication Type: Journal article, Original article
Publication year: 2014
Journal
Publisher: University of Münster
Pages Range: 463-487
Journal Issue: 7
DOI: 10.17879/58269762646
Abstract
The notion of a linear Coxeter system introduced by Vinberg generalizes the geometric representation of a Coxeter group. Our main theorem asserts that if v is an element of the Tits cone of a linear Coxeter system and $\cW$ is the corresponding Coxeter group, then $\cW v \subeq v - C_v,$ where Cv is the convex cone generated by the coroots αˇ, for which α(v)>0. This implies that the convex hull of $\cW v$ is completely determined by the image of v under the reflections in $\cW$. We also apply an analogous result for convex hulls of $\cW$-orbits in the dual space, although this action need not correspond to a linear Coxeter system. Motivated by the applications in representation theory, we further extend these results to Weyl group orbits of locally finite and locally affine root systems. In the locally affine case, we also derive some applications on minimizing linear functionals on Weyl group orbits.
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How to cite
APA:
Hofmann, G., & Neeb, K.H. (2014). On convex hulls of orbits of Coxeter groups and Weyl groups. Münster Journal of Mathematics, 7, 463-487. https://doi.org/10.17879/58269762646
MLA:
Hofmann, Georg, and Karl Hermann Neeb. "On convex hulls of orbits of Coxeter groups and Weyl groups." Münster Journal of Mathematics 7 (2014): 463-487.
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