Syzygy Algebras for Segre Embeddings

We describe the syzygy spaces for the Segre embedding $\mathbb{P}(U)\times\mathbb{P}(V)\subset\mathbb{P}(U\otimes V)$ in terms of representations of $\operatorname{GL}(U)\times\operatorname{GL}(V)$ and construct the minimal resolutions of the sheaves $\mathcal{O}_{\mathbb{P}(U)\times\mathbb{P}(V)}(a,b)$ in $D(\mathbb{P}(U\otimes V))$ for $a\ge-\dim(U)$ and $b\ge-\dim(V)$. We also prove a property of multiplication in syzygy spaces of the Segre embedding.