Schubert decompositions for quiver Grassmannians of tree modules

Lorscheid, Oliver

doi:10.2140/ant.2015.9.1337

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Schubert decompositions for quiver Grassmannians of tree modules

Oliver Lorscheid

Appendix: Thorsten Weist

Vol. 9 (2015), No. 6, 1337–1362

DOI: 10.2140/ant.2015.9.1337

Abstract

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

In particular, we determine the Euler characteristic of Gr $_{\underline{e}} (M)$ as the number of extremal successor closed subsets of $T_{0}$ , which extends the results of Cerulli Irelli (2011) and Haupt (2012) (under certain additional assumptions on $ℬ$ ).

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

Vol. 9, No. 6, 2015