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Reflection groups, reflection arrangements, and invariant real varieties

Authors: Tobias Friedl, Cordian Riener and Raman Sanyal
Journal: Proc. Amer. Math. Soc. 146 (2018), 1031-1045
MSC (2010): Primary 14P05, 14P10, 20F55
Published electronically: October 18, 2017
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Abstract: Let $ X$ be a nonempty real variety that is invariant under the action of a reflection group $ G$. We conjecture that if $ X$ is defined in terms of the first $ k$ basic invariants of $ G$ (ordered by degree), then $ X$ meets a $ k$-dimensional flat of the associated reflection arrangement. We prove this conjecture for the infinite types, reflection groups of rank at most $ 3$, and $ F_4$ and we give computational evidence for $ H_4$. This is a generalization of Timofte's degree principle to reflection groups. For general reflection groups, we compute nontrivial upper bounds on the minimal dimension of flats of the reflection arrangement meeting $ X$ from the combinatorics of parabolic subgroups. We also give generalizations to real varieties invariant under Lie groups.

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Additional Information

Tobias Friedl
Affiliation: Fachbereich Mathematik und Informatik, Freie Universität Berlin, 14195 Berlin, Germany

Cordian Riener
Affiliation: Aalto Science Institute, P.O. Box 11000, FI-00076 Aalto, Finland

Raman Sanyal
Affiliation: Institut für Mathematik, Goethe-Universität Frankfurt, 60325 Frankfurt, Germany

Keywords: Reflection groups, reflection arrangements, invariant real varieties, real orbit spaces
Received by editor(s): November 14, 2016
Received by editor(s) in revised form: May 11, 2017
Published electronically: October 18, 2017
Additional Notes: The first and third authors were supported by the DFG-Collaborative Research Center, TRR 109 “Discretization in Geometry and Dynamics”. The first author received additional funding from a scholarship of the Dahlem Research School at Freie Universität Berlin.
Communicated by: Patricia Hersh
Article copyright: © Copyright 2017 American Mathematical Society

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