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Quiver Grassmannians of Extended Dynkin type $D$ Part 1: Schubert Systems and Decompositions Into Affine Spaces
About this Title
Oliver Lorscheid, Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, Brazil and Thorsten Weist, Bergische Universität Wuppertal, Gaußstr. 20, 42097 Wuppertal, Germany
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 261, Number 1258
ISBNs: 978-1-4704-3647-6 (print); 978-1-4704-5399-2 (online)
DOI: https://doi.org/10.1090/memo/1258
Published electronically: November 12, 2019
MSC: Primary 13F60, 14F45, 14M15, 14N15, 16G20; Secondary 05E10, 14M17, 16G60
Table of Contents
Chapters
- Introduction
- 1. Background
- 2. Schubert systems
- 3. First applications
- 4. Schubert decompositions for type $\widetilde D_n$
- 5. Proof of Theorem 4.1
- A. Representations for quivers of type $\widetilde D_n$
- B. Bases for representations of type $\widetilde D_n$
Abstract
Let $Q$ be a quiver of extended Dynkin type $\widetilde D_n$. In this first of two papers, we show that the quiver Grassmannian $Gr_{\underline {e}}(M)$ has a decomposition into affine spaces for every dimension vector ${\underline {e}}$ and every indecomposable representation $M$ of defect $-1$ and defect $0$, with exception of the non-Schurian representations in homogeneous tubes. We characterize the affine spaces in terms of the combinatorics of a fixed coefficient quiver for $M$. The method of proof is to exhibit explicit equations for the Schubert cells of $Gr_{\underline {e}}(M)$ and to solve this system of equations successively in linear terms. This leads to an intricate combinatorial problem, for whose solution we develop the theory of Schubert systems.
In Part 2 of this pair of papers, we extend the result of this paper to all indecomposable representations $M$ of $Q$ and determine explicit formulae for the $F$-polynomial of $M$.
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