# Wydawnictwa / Czasopisma IMPAN / Colloquium Mathematicum / Wszystkie zeszyty

## Schubert varieties and representations of Dynkin quivers

### Tom 94 / 2002

Colloquium Mathematicum 94 (2002), 285-309 MSC: 14B05, 14L30, 14M15, 16G20, 16G70. DOI: 10.4064/cm94-2-10

#### Streszczenie

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.

#### Autorzy

• Grzegorz BobińskiFaculty of Mathematics and Computer Science
Nicholas Copernicus University
Chopina 12/18
87-100 Toruń, Poland
e-mail
• Grzegorz ZwaraFaculty of Mathematics and Computer Science
Nicholas Copernicus University
Chopina 12/18
87-100 Toruń, Poland
e-mail

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