Characterizations of strong semilinear embeddings in terms of general linear and projective linear groups

TL;DR

This paper characterizes strong semilinear embeddings between vector spaces over division rings using their interaction with general linear and projective linear groups, providing conditions under which such embeddings are uniquely determined.

Contribution

It establishes new criteria for when a map is a strong semilinear embedding based on group actions, extending previous results and including specific examples and conditions.

Findings

A map is a strong semilinear embedding if it respects group actions and is non-constant.

Conditions on the image dimension are necessary for the characterization.

The results connect group actions with geometric embeddings in vector spaces.

Abstract

Let $V$ and $V'$ be vector spaces over division rings. Suppose $\dim V$ is finite and not less than 3. Consider a mapping $l:V\to V$ with the following property: for every $u\in {\rm GL}(V)$ there is $u'\in {\rm GL}(V')$ such that $lu=u'l$. Our first result states that $l$ is a strong semilinear embedding if $l|_{V\setminus{0}}$ is non-constant and the dimension of the subspace of $V'$ spanned by $l(V)$ is not greater than $n$. We present examples showing that these conditions can not be omitted. In some special cases, this statement can be obtained from Dicks and Hartley (1991) and Zha (1996). Denote by ${\mathcal P}(V)$ the projective space associated with $V$ and consider the mapping $f:{\mathcal P}(V)\to {\mathcal P}(V')$ with the following property: for every $h\in {\rm PGL}(V)$ there is $h'\in {\rm PGL}(V')$ such that $fh=h'f$. By the second result, $f$ is induced by a strong semilinear embedding of $V$ in $V'$ if $f$ is non-constant and its image is contained in a subspace of $V'$ whose dimension is not greater than $n$, we also require that $R'$ is a field.