Affine Schubert Varieties and Circular Complexes

Abstract

Schubert varieties have been exhaustively studied with a plethora of techniques: Coxeter groups, explicit desingularization, Frobenius splitting, etc. Many authors have applied these techniques to various other varieties, usually defined by determinantal equations. It has turned out that most of these apparently different varieties are actually Schubert varieties in disguise, so that one may use a single unified theory to understand many large families of spaces. The most powerful result in this direction was given by Lusztig as a footnote to his work on canonical bases. He showed that the variety of nilpotent representations of a cyclic quiver (including nilpotent conjugacy classes of matrices) is isomorphic to an open subset of a Schubert variety for the loop group GL(n)^. In this paper, we attempt to describe the affine Schubert varieties and Lusztig's isomorphism in the simplest terms possible. We then apply this isomorphism to an interesting example, the variety of circular complexes, recovering some results of Mehta and Trivedi.