Overgroups of exterior powers of an elementary group. levels

ABSTRACT

We prove a first part of the standard description of groups H lying between an exterior power of an elementary group $m{\mathrm{E}}_n\left(R\right)$ and a general linear group ${\mathrm{GL}}_{(\genfrac{}{}{0ex}{}{n}{m})}\left(R\right)$ for a commutative ring $R,2\in R^{\ast }$ and $n⩾3m$. The description uses the classical notion of a level: for every group H we find a unique ideal A of the ground ring R, which describes H.

KEYWORDS:

• General linear group
• elementary group
• overgroup
• fundamental representation
• exterior power
• level

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20G35

Acknowledgements

We would like to express our sincere gratitude to our scientific adviser Nikolai Vavilov for formulating the problem and for a constant support, without which this paper would never have been written. The authors are grateful to Alexei Stepanov for carefully reading our original manuscript and for numerous remarks and corrections. Also, we would like to thank an anonymous referee for bringing our attention to the paper [46].

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The restriction of the exterior square map $2:{\mathrm{GL}}_4\left(R\right)⟶{\mathrm{GL}}_6\left(R\right)$ to the group ${\mathrm{E}}_4\left(R\right)$ is an isomorphism onto the elementary orthogonal group ${\mathrm{EO}}_6\left(R\right)$ [24].

2 The same strict inclusions are still true with changing $\mathrm{GL}$ to $\mathrm{SL}$.

3 Recall that we consider the representation with the highest weight ${\varpi }_m$.

4 From root systems geometry any vertex can be initial or terminal for at most one α-path.

Additional information

Funding

This work was supported by Russian Science Foundation [grant number 17-11-01261].