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Range-compatible homomorphisms on spaces of symmetric or alternating matrices

Let $U$ and $V$ be finite-dimensional vector spaces over an arbitrary field $\mathbb{K}$, and $\mathcal{S}$ be a linear subspace of the space $\mathcal{L}(U,V)$ of all linear maps from $U$ to $V$. A map $F : \mathcal{S} \rightarrow V$ is called range-compatible when it satisfies $F(s) \in \mathrm{im}(s)$ for all $s \in \mathcal{S}$. Among the range-compatible maps are the so-called local ones, that is the maps of the form $s \mapsto s(x)$ for a fixed vector $x$ of $U$. In recent works, we have classified the range-compatible group homomorphisms on $\mathcal{S}$ when the codimension of $\mathcal{S}$ in $\mathcal{L}(U,V)$ is small. In the present article, we study the special case when $\mathcal{S}$ is a linear subspace of the space $S_n(\mathbb{K})$ of all $n$ by $n$ symmetric matrices: we prove that if the codimension of $\mathcal{S}$ in $S_n(\mathbb{K})$ is less than or equal to $n-2$, then every range-compatible homomorphism on $\mathcal{S}$ is local provided that $\mathbb{K}$ does not have characteristic $2$. With the same assumption on the codimension of $\mathcal{S}$, we also classify the range-compatible homomorphisms on $\mathcal{S}$ when $\mathbb{K}$ has characteristic $2$. Finally, we prove that if $\mathcal{S}$ is a linear subspace of the space $A_n(\mathbb{K})$ of all $n$ by $n$ alternating matrices with entries in $\mathbb{K}$, and the codimension of $\mathcal{S}$ is less than or equal to $n-3$, then every range-compatible homomorphism on $\mathcal{S}$ is local.