This question might be too elementary but it arises naturally as a part of a more complicated computation and I struggle to find the answer.

Let $V$ be an $n$-dimensional complex vector space. Consider a map $$Sym^m(V)\otimes Sym^k(V)\longrightarrow Sym^{m-1}(V)\otimes Sym^{k+1}(V)$$ given by $$x_1 \dots x_m \otimes y_1 \dots y_k \mapsto \sum x_1 \dots x_{i-1} x_{i+1} \dots x_m \otimes x_i y_1 \dots y_k.$$ For $m=k=1$ this map is surjective and for $m=2,\ k=0$ it is injective. In the first case the kernel is $\Lambda^2 V$ which is also a cokernel for the second case. For $n=1$ it is always an isomorphism.

- For what triples of $n, m, k$ this map is injective or surjective?
- Can the kernel / cokernel be presented functorially in terms of V?
- Is this map a part of some natural resolution?