Reflection groups, reflection arrangements, and invariant real varieties

Tobias Friedl; Cordian Riener; Raman Sanyal

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arXiv · 1602.06732

Reflection groups, reflection arrangements, and invariant real varieties

Tobias Friedl·Cordian Riener·Raman Sanyal

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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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Tobias Friedl, Cordian Riener, Raman Sanyal. 2016-02-22. Reflection groups, reflection arrangements, and invariant real varieties. https://arxiv.org/abs/1602.06732

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