Orbits of in dimension 8

Abstract

Groups of structure $2.O_8^{+}(2)$ have an irreducible representation of degree 8 which can be realized over $\mathbb{Z}$ and any prime field ${\mathbb{F}}_p.$ This representation extends to a group of structure $2.O_8^{+}(2).2.$ Any subgroup $Z\le {\mathbb{F}}_p^{\times }$ acts by scalar multiplication on this module over ${\mathbb{F}}_p.$ In this short note we determine for which primes p > 7 and which Z the central products $Z°(2.O_8^{+}(2)$ and $Z°(2.O_8^{+}(2).2)$ have a regular orbit on the 8-dimensional ${\mathbb{F}}_p$-module. This work was triggered by an omission in the paper [4] by Köhler and Pahlings, a paper which is used in various places in work on the k(GV)-problem.

Keywords:

• regular orbits
• Weyl group of type E₈

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20C20
• 20C40
• 20F55

Acknowledgment

I thank Melissa Lee for making me aware of the gap concerning $O_8^{+}(2)$ in the paper [4]; this caused a corresponding gap in early versions of her article [5] (which hopefully can now be closed).

Additional information

Funding

This is a contribution to Project-ID 286237555 – TRR 195 – by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).