  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 ${\rm GL}(U)\times {\rm GL}(V)$ and construct the minimal
resolutions of the sheaves
$\mathscr{O}_{\mathbb{P}(U)\times\mathbb{P}(V)}(a,b)$ in $D(\mathbb{P}(U\otimes
V))$ for $a\geqslant-\dim(U)$ and $b\geqslant-\dim(V)$. Also we prove some
property of multiplication on syzygy spaces of the Segre embedding.
