Degenerate Affine Flag Varieties and Quiver Grassmannians
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Degenerate Affine Flag Varieties and Quiver Grassmannians ¨ 1 Alexander Putz Received: 1 April 2020 / Accepted: 16 November 2020 / © The Author(s) 2020
Abstract We study finite dimensional approximations to degenerate versions of affine flag varieties using quiver Grassmannians for cyclic quivers. We prove that they admit cellular decompositions parametrized by affine Dellac configurations, and that their irreducible components are normal Cohen-Macaulay varieties with rational singularities. Keywords Linear degenerations · Finite approximations · Equioriented cycle · Rational singularities · Grand Motzkin paths · Affine Dellac configurations Mathematics Subject Classification (2010) 16G20 · 14M15 · 14D06 · 17B67
1 Introduction This work is based on the identification of the degenerate type A flag variety with a quiver Grassmannian for the equioriented quiver of type A as shown by G. Cerulli Irelli, E. Feigin and M. Reineke in [7]. Additionally, it is based on a similar construction, using quiver Grassmannians for the loop quiver, giving finite approximations of the degenerate affine Grassmannian as introduced by E. Feigin, M. Finkelberg and M. Reineke in [14]. We generalise their constructions to describe finite approximations of degenerate affine flag varieties, using quiver Grassmannians for the equioriented cycle. Linear degenerations of the affine flag variety are defined similarly to the construction for the type A flag variety by G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier and M. Reineke [6]. Quiver Grassmannians were first used by W. Crawley-Boevey and A. Schofield [8, 31]. In some special cases quiver Grassmannians for the equioriented cycle were studied by N. Haupt [18, 19]. The variety of representations of the cycle was studied by G. Kempken [23]. J. Sauter studied the quiver flag variety for the equioriented cycle [29]. The Ringel-Hall algebra of the cyclic quiver was studied by A. Hubery [20]. Presented by: Peter Littelmann Alexander P¨utz
[email protected] 1
Faculty of Mathematics, Ruhr-University Bochum, Universit¨atsstrasse 150, 44780 Bochum, Germany
¨ A. Putz
Based on the work by G. Kempken and A. Hubery we derive statements about the geometry of quiver Grassmannians for the equioriented cycle and generalise a result by N. Haupt about the parametrisation of their irreducible components. The main goal of this paper is the study of degenerate affine flag varieties of type gln . Analogous to the classical setting the affine flag variety is defined as the quotient of the affine Kac-Moody group by its standard Iwahori subgroup [25, Chapter XIII]. Based on the identification of the finite approximations with quiver Grassmannians for the equioriented cycle, we examine geometric properties of the degenerations.
1.1 Main Results Finite approximations of the degenerate affine flag variety are isomorphic to certain quiver Grassmannians for the cyclic quiver, as shown in Theorem 3.7. The approximations have equidimensional irreducible components, which are parametrised by grand Motzkin paths,
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