Minimum codimension of eigenspaces in irreducible representations of simple classical linear algebraic groups

Abstract

Let k be an algebraically closed field of characteristic $p\ge 0$, let G be a simple simply connected classical linear algebraic group of rank $\mathcal{l}$ and let T be a maximal torus in G with rational character group $\mathrm{X}(T)$. For a nonzero p-restricted dominant weight $\lambda \in \mathrm{X}(T)$, let V be the associated irreducible kG-module. We define ${\nu }_G(V)$ as the minimum codimension of any eigenspace on V for any non-central element of G. In this paper, we determine lower-bounds for ${\nu }_G(V)$ for G of type $A_{\mathcal{l}}$ and $\dim (V)\le \frac{{\mathcal{l}}^3}{2}$, and for G of type $B_{\mathcal{l}},C_{\mathcal{l}}$, or $D_{\mathcal{l}}$ and $\dim (V)\le 4{\mathcal{l}}^3$. Moreover, we give the exact value of ${\nu }_G(V)$ for G of type $A_{\mathcal{l}}$ with $\mathcal{l}\ge 15$; for G of type $B_{\mathcal{l}}$ or $C_{\mathcal{l}}$ with $\mathcal{l}\ge 14$; and for G of type $D_{\mathcal{l}}$ with $\mathcal{l}\ge 16$.

KEYWORDS:

• Linear algebraic groups
• representation theory

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20G05

Acknowledgments

The author is immensely grateful to Donna Testerman for her guidance. The author would also like to thank Simon Goodwin, Martin Liebeck and Adam Thomas for many helpful discussions and comments.

Disclosure statement

The author reports there are no competing interests to declare.

Additional information

Funding

This work was supported by the Swiss National Science Foundation, grant number FNS 200020 175571, and by the Engineering and Physical Sciences Research Council, grant number EP/R018952/1.