Vol. 6, No. 1, 2012

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Quiver Grassmannians and degenerate flag varieties

Giovanni Cerulli Irelli, Evgeny Feigin and Markus Reineke

Vol. 6 (2012), No. 1, 165–194
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