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

TL;DR

This paper studies power maps in the group of invertible matrices over finite local rings of length two, classifies elements based on their properties, and provides explicit probability generating functions for these classifications.

Contribution

It introduces a novel analysis of power maps in $ ext{GL}_n( ext{ring})$ over finite principal ideal local rings of length two, including canonical forms and probability calculations.

Findings

Classified elements in the image of power maps with specific properties.

Established a Hensel lifting technique for matrix polynomial equations.

Derived explicit probability generating functions for various matrix properties.

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 $\mathrm{GL}_n(\mathscr O_2)$, where $\mathscr O_2$ is a finite local principal ideal ring of length two (e.g. $\mathbb{Z}/p^2\mathbb{Z}$ and $\mathbb F_q[t]/\langle t^2\rangle$). 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 $\mathscr O_2$. 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 $\mathscr O_2$. Our main tool is a Hensel lifting for polynomial equations over $\mathrm{M}_n(\mathscr O_2)$, 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 $\mathscr O_2$. As applications, we derive explicit generating functions for the probabilities that a random element of $\mathrm{GL}_n(\mathscr O_2)$ is regular semisimple, $L$-power regular semisimple, compatible cyclic, or $L$-power compatible cyclic.