Unipotent orbits of elements in a quaternion ring of odd order

TL;DR

This paper classifies unipotent similarity classes in quaternion rings over finite local rings of odd order by analyzing conjugation actions and reducing the problem to matrix rings over the base ring.

Contribution

It provides a classification of unipotent orbits in quaternion rings over finite local rings, extending understanding of their conjugation actions and similarity classes.

Findings

Classification of unipotent similarity classes in quaternion rings.

Reduction of the problem to matrix rings over the base ring.

Explicit description of conjugation actions on unipotent elements.

Abstract

Let $n \in \NN$ and let $q=p^r$ be an odd prime power. Let $R$ be a finite commutative local principal ring of cardinality $q^{n}$ with $R/J(R) \simeq GF(q)$. We study the conjugation action of the group of all unipotent elements in the quaternion ring $H(R)$ on $H(R)$ and we classify the resulting unipotent similarity classes, using a reduction to the ring of $2$-by-$2$ matrices over $R$.