Quiver Grassmannians and degenerate flag varieties

Cerulli Irelli, Giovanni; Feigin, Evgeny; Reineke, Markus

doi:10.2140/ant.2012.6.165

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Quiver Grassmannians and degenerate flag varieties

Giovanni Cerulli Irelli, Evgeny Feigin and Markus Reineke

Vol. 6 (2012), No. 1, 165–194

DOI: 10.2140/ant.2012.6.165

Abstract

Quiver Grassmannians are varieties parametrizing subrepresentations of a quiver representation. It is observed that certain quiver Grassmannians for type A quivers are isomorphic to the degenerate flag varieties investigated earlier by Feigin. This leads to the consideration of a class of Grassmannians of subrepresentations of the direct sum of a projective and an injective representation of a Dynkin quiver. It is proved that these are (typically singular) irreducible normal local complete intersection varieties, which admit a group action with finitely many orbits and a cellular decomposition. For type A quivers, explicit formulas for the Euler characteristic (the median Genocchi numbers) and the Poincaré polynomials are derived.

Keywords

flag variety, quiver grassmannian, degeneration

Mathematical Subject Classification 2010

Primary: 14M15

Secondary: 16G20

Milestones

Received: 16 June 2011

Revised: 18 July 2011

Accepted: 14 August 2011

Published: 15 June 2012

Authors

Giovanni Cerulli Irelli
Sapienza – Università di Roma Piazzale Aldo Moro 5 00185 Rome Italy

Evgeny Feigin
Department of Mathematics National Research University Higher School of Economics Vavilova str. 7 Moscow 117312 Russia

Markus Reineke
Fachbereich C - Mathematik Bergische Universität Wuppertal, D- 42097 Wuppertal Germany

Vol. 6, No. 1, 2012