Quiver Grassmannians of Extended Dynkin Type $D$

Part I: Schubert Systems and Decompositions Into Affine Spaces

Oliver Lorscheid, Thorsten Weist (Autoren)

Buch | Softcover

80 Seiten

2019
American Mathematical Society (Verlag)
978-1-4704-3647-6 (ISBN)

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Let $Q$ 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 $M$.

Oliver Lorscheid, Instituto Nacional de Matematica Pura e Aplicada, Rio de Janeiro, Brazil. Thorsten Weist, Bergische Universitat Wuppertal, Germany.

Introduction
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.

Erscheinungsdatum	31.12.2019
Reihe/Serie	Memoirs of the American Mathematical Society
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	180 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	1-4704-3647-7 / 1470436477
ISBN-13	978-1-4704-3647-6 / 9781470436476
Zustand	Neuware