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OverviewLet $Q$ be a quiver of extended Dynkin type $\widetilde{D}_n$. In this first of two papers, the authors show that the quiver Grassmannian $\mathrm{Gr}_{\underline{e}}(M)$ has a decomposition into affine spaces for every dimension vector $\underline{e}$ and every indecomposable representation $M$ of defect $-1$ and defect $0$, with the exception of the non-Schurian representations in homogeneous tubes. The authors 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 $\mathrm{Gr}_{\underline{e}}(M)$ and to solve this system of equations successively in linear terms. This leads to an intricate combinatorial problem, for whose solution the authors develop the theory of Schubert systems. In Part 2 of this pair of papers, they extend the result of this paper to all indecomposable representations $M$ of $Q$ and determine explicit formulae for the $F$-polynomial of $M$. Full Product DetailsAuthor: Oliver Lorscheid , Thorsten WeistPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.180kg ISBN: 9781470436476ISBN 10: 1470436477 Pages: 80 Publication Date: 30 December 2019 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsIntroduction Background Schubert systems First applications Schubert decompositions for type $\widetilde{D}_n$ Proof of Theorem 4.1 Appendix A. Representations for quivers of type $\widetilde{D}_n$ Appendix B. Bases for representations of type $\widetilde{D}_n$ Bibliography.ReviewsAuthor InformationOliver Lorscheid, Instituto Nacional de Matematica Pura e Aplicada, Rio de Janeiro, Brazil. Thorsten Weist, Bergische Universitat Wuppertal, Germany. Tab Content 6Author Website:Countries AvailableAll regions |
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