Vol. 9, No. 6, 2015

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Schubert decompositions for quiver Grassmannians of tree modules

Oliver Lorscheid

Appendix: Thorsten Weist

Vol. 9 (2015), No. 6, 1337–1362
Abstract

Let Q be a quiver, M a representation of Q with an ordered basis and e¯ a dimension vector for Q. In this note we extend the methods of Lorscheid (2014) to establish Schubert decompositions of quiver Grassmannians Gre¯(M) into affine spaces to the ramified case, i.e., the canonical morphism F : T Q from the coefficient quiver T of M w.r.t.  is not necessarily unramified.

In particular, we determine the Euler characteristic of Gre¯(M) as the number of extremal successor closed subsets of T0, which extends the results of Cerulli Irelli (2011) and Haupt (2012) (under certain additional assumptions on ).

Keywords
quiver Grassmannian, Schubert decompositions, Euler characteristics, tree modules
Mathematical Subject Classification 2010
Primary: 14M15
Secondary: 16G20, 05C05
Milestones
Received: 22 August 2013
Revised: 23 February 2015
Accepted: 17 June 2015
Published: 7 September 2015
Authors
Oliver Lorscheid
Instituto Nacional de Matemática Pura e Aplicada
Estrada Dona Castorina 110
22460-320 Rio de Janeiro-RJ
Brazil
Thorsten Weist
Bergische Universität Wuppertal
Gaussstr. 20
42097 Wuppertal
Germany