Syzygies of quadratic Veronese embedding

In this work we explicitly calculate the syzygies of the quadratic Veronese embedding $\mathbb{P}(V)\subset\mathbb{P}(\operatorname{Sym}^2V)$ as representations of the group $\operatorname{GL}(V)$. Resolutions of the sheaves $\mathscr{O}_{\mathbb{P}(V)}(i)$ are also constructed in the category $D(\mathbb{P}(\operatorname{Sym}^2V))$.
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