Schubert varieties and representations of Dynkin quivers

Grzegorz Bobiński¹ ; Grzegorz Zwara¹

¹ Faculty of Mathematics and Computer Science Nicholas Copernicus University Chopina 12/18 87-100 Toruń, Poland

Colloquium Mathematicum, Tome 94 (2002) no. 2, pp. 285-309

Résumé

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.

Détail

DOI : 10.4064/cm94-2-10

Keywords: types singularities schubert varieties flag varieties mathop flag nolimits mathbb equivalent types singularities orbit closures representations dynkin quivers type mathbb similarly prove types singularities schubert varieties products grassmannians mathop grass nolimits times mathop grass nolimits mathbb leq equivalent types singularities orbit closures representations dynkin quivers type mathbb orbit closures representation varieties dynkin quivers type mathbb normal cohen macaulay varieties