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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Reflection groups, reflection arrangements, and invariant real varieties
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by Tobias Friedl, Cordian Riener and Raman Sanyal PDF
Proc. Amer. Math. Soc. 146 (2018), 1031-1045 Request permission


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
  • Email:
  • Cordian Riener
  • Affiliation: Aalto Science Institute, P.O. Box 11000, FI-00076 Aalto, Finland
  • MR Author ID: 816514
  • Email:
  • Raman Sanyal
  • Affiliation: Institut für Mathematik, Goethe-Universität Frankfurt, 60325 Frankfurt, Germany
  • MR Author ID: 856938
  • Email:
  • 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
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 146 (2018), 1031-1045
  • MSC (2010): Primary 14P05, 14P10, 20F55
  • DOI:
  • MathSciNet review: 3750216