## 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

## 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.## References

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## Additional Information

**Tobias Friedl**- Affiliation: Fachbereich Mathematik und Informatik, Freie Universität Berlin, 14195 Berlin, Germany
- Email: tfriedl@zedat.fu-berlin.de
**Cordian Riener**- Affiliation: Aalto Science Institute, P.O. Box 11000, FI-00076 Aalto, Finland
- MR Author ID: 816514
- Email: cordian.riener@aalto.fi
**Raman Sanyal**- Affiliation: Institut für Mathematik, Goethe-Universität Frankfurt, 60325 Frankfurt, Germany
- MR Author ID: 856938
- Email: sanyal@math.uni-frankfurt.de
- 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: https://doi.org/10.1090/proc/13821
- MathSciNet review: 3750216