Schubert varieties and representations of Dynkin quivers
Volume 94 / 2002
Abstract
We show that the types of singularities of Schubert varieties in the flag varieties $\mathop{\rm Flag}\nolimits_n$, $n \in {\mathbb N}$, are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type $\mathbb A$. Similarly, we prove that the types of singularities of Schubert varieties in products of Grassmannians $\mathop{\rm Grass}\nolimits (n, a) \times \mathop{\rm Grass}\nolimits (n, b)$, $a, b, n \in {\mathbb N}$, $a, b \leq n$, are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type $\mathbb D$. We also show that the orbit closures in representation varieties of Dynkin quivers of type ${\mathbb D}$ are normal and Cohen–Macaulay varieties.
