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## Syzygy algebras for the 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 $\GL(U)\times \GL(V)$ 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.