Tangent Spaces of Orbit Closures for Representations of Dynkin Quivers of Type $\mathbb {D}$

Abstract

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