Power maps on General Linear groups over finite principal ideal local rings of length two

Abstract

Word maps have been studied for matrix groups over a field. We initiate the study of problems related to word maps in the context of the group ${\mathrm{GL}}_n({\mathcal{O}}_2)$, where ${\mathcal{O}}_2$ is a finite local principal ideal ring of length two (e.g. $\mathbb{Z}/p^2\mathbb{Z}$ and ${\mathbb{F}}_q[t]/⟨t^2⟩$). We study the power map $g\mapsto g^L$, where $L$ is a positive integer. We consider $L$ to be coprime to $p$ (an odd prime), the characteristic of the residue field $k$ of ${\mathcal{O}}_2$. We classify all the elements in the image, whose mod-$\mathfrak{m}$ reduction in ${\mathrm{GL}}_n(k)$ are either regular semisimple or cyclic, where $\mathfrak{m}$ is the unique maximal ideal of ${\mathcal{O}}_2$. Our main tool is a Hensel lifting for polynomial equations over ${\mathrm{M}}_n({\mathcal{O}}_2)$, which we establish in this work. A central contribution of this work is the construction of canonical forms for certain natural classes of matrices over ${\mathcal{O}}_2$. As applications, we derive explicit generating functions for the probabilities that a random element of ${\mathrm{GL}}_n({\mathcal{O}}_2)$ is regular semisimple, $L$-power regular semisimple, compatible cyclic, or $L$-power compatible cyclic.

32 pages; 7 examples; Preliminary version; Comments are always welcome!