We show that the types of singularities of Schubert varieties in the flag varieties Flagₙ, n ∈ ℕ, are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type 𝔸. Similarly, we prove that the types of singularities of Schubert varieties in products of Grassmannians Grass(n,a) × Grass(n,b), a, b, n ∈ ℕ, a, b ≤ n, are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type 𝔻. We also show that the orbit closures in representation varieties of Dynkin quivers of type 𝔻 are normal and Cohen-Macaulay varieties.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm94-2-10,
author = {Grzegorz Bobi\'nski and Grzegorz Zwara},
title = {Schubert varieties and representations of Dynkin quivers},
journal = {Colloquium Mathematicae},
volume = {91},
year = {2002},
pages = {285-309},
zbl = {1013.14011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm94-2-10}
}
Grzegorz Bobiński; Grzegorz Zwara. Schubert varieties and representations of Dynkin quivers. Colloquium Mathematicae, Tome 91 (2002) pp. 285-309. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm94-2-10/