Tangent spaces of orbit closures for representations of Dynkin quivers of type D

Abstract

Let $\mathbb{k}$ be an algebraically closed field, $Q$ a finite quiver, and denote by ${\mathrm{rep}}_Q^{\mathbf{d}}$ the affine $\mathbb{k}$-scheme of representations of $Q$ with a fixed dimension vector $\mathbf{d}$. Given a representation $M$ of $Q$ with dimension vector $\mathbf{d}$, the set ${\mathcal{O}}_M$ of points in ${\mathrm{rep}}_Q^{\mathbf{d}}(\mathbb{k})$ isomorphic as representations to $M$ is an orbit under an action on ${\mathrm{rep}}_Q^{\mathbf{d}}(\mathbb{k})$ of a product of general linear groups. The orbit ${\mathcal{O}}_M$ and its Zariski closure ${\overline{\mathcal{O}}}_M$, considered as reduced subschemes of ${\mathrm{rep}}_Q^{\mathbf{d}}$, are contained in an affine scheme ${\mathcal{C}}_M$ defined by rank conditions on suitable matrices associated to ${\mathrm{rep}}_Q^{\mathbf{d}}$. For all Dynkin and extended Dynkin quivers, the sets of points of ${\overline{\mathcal{O}}}_M$ and ${\mathcal{C}}_M$ coincide, or equivalently, ${\overline{\mathcal{O}}}_M$ is the reduced scheme associated to ${\mathcal{C}}_M$. Moreover, ${\overline{\mathcal{O}}}_M={\mathcal{C}}_M$ provided $Q$ is a Dynkin quiver of type $\mathbb{A}$, and this equality is a conjecture for the remaining Dynkin quivers (of type $\mathbb{D}$ and $\mathbb{E}$). Let $Q$ be a Dynkin quiver of type $\mathbb{D}$ and $M$ a finite dimensional representation of $Q$. We show that the equality $T_N{\overline{\mathcal{O}}}_M=T_N{\mathcal{C}}_M$ of Zariski tangent spaces holds for any closed point $N$ of ${\overline{\mathcal{O}}}_M$. As a consequence, we describe the tangent spaces to ${\overline{\mathcal{O}}}_M$ in representation theoretic terms.