Range-compatible homomorphisms over fields with two elements

Abstract

Let $U$ and $V$ be finite-dimensional vector spaces over a (commutative) field $\mathbb{K}$, and $\mathcal{S}$ be a linear subspace of the space $\mathcal{L}(U,V)$ of all linear operators from $U$ to $V$. A map $F:\mathcal{S}\to V$ is called range-compatible when $F(s)\in \text{Im}(s)$ for all $s\in \mathcal{S}$. In a previous work, we have classified all the range-compatible group homomorphisms provided that $\text{codim}(\mathcal{S})\le 2\text{dim}(V)-3$, except in the special case when $\mathbb{K}$ has only two elements and $\text{codim}(\mathcal{S})=2\text{dim}(V)-3$. In this article, we give a thorough treatment of that special case. Our results are partly based upon the recent classification of vector spaces of matrices with rank at most $2$ over ${\mathbb{F}}_2$. As an application, we classify the $2$-dimensional non-reflexive operator spaces over any field, and the affine subspaces of ${\text{M}}_{n,p}(\mathbb{K})$ with lower-rank $2$ and codimension $3$.

64 pages

Cited by in corpus (1)

• Quasi-range-compatible affine maps on large operator spaces