TY  - BOOK
AU  - Lorscheid,Oliver
AU  - Weist,Thorsten
TI  - Quiver Grassmannians of Extended Dynkin Type : Schubert Systems and Decompositions into Affine Spaces
T2  - Memoirs of the American Mathematical Society Series
SN  - 9781470453992
AV  - QA613.6 .L67 2019
PY  - 2019///
CY  - Providence
PB  - American Mathematical Society
KW  - Dynkin diagrams
KW  - Grassmann manifolds
KW  - Mathematics
KW  - Electronic books
N1  - Cover -- Title page -- Introduction -- Chapter 1. Background -- 1.1. Coefficient quiver -- 1.2. Schubert decompositions -- 1.3. Representations of Schubert cells -- Chapter 2. Schubert systems -- 2.1. The complete Schubert system -- 2.2. Partial Evaluations -- 2.3. Contradictory  -states -- 2.4. Definition of  -states -- 2.5. The reduced Schubert system -- 2.6. Computing  -states -- 2.7. Solvable  -states -- 2.8. Extremal edges -- 2.9. Patchwork solutions -- 2.10. Extremal paths -- Chapter 3. First applications -- 3.1. The Kronecker quiver -- 3.2. Dynkin quivers -- Chapter 4. Schubert decompositions for type ̃ _{ } -- 4.1. Contradictory   of the first and of the second kind -- 4.2. Automorphisms of the Dynkin diagram -- 4.3. Bases for some indecomposable representations -- 4.4. The main theorem -- Chapter 5. Proof of Theorem 4.1 -- 5.1. Defect -1 -- Appendix A. Representations for quivers of type ̃ _{ } -- A.1. Reflections and Auslander-Reiten translates -- A.2. Indecomposable and exceptional representations -- A.3. The Auslander-Reiten quiver -- A.4. The tubes -- A.5. Roots -- A.6. The defect -- Appendix B. Bases for representations of type ̃ _{ } -- B.1. Defect -1 -- B.2. Defect -2 -- B.3. Positive defect -- B.4. Exceptional tubes of rank 2 -- B.5. Exceptional tubes of rank  -2 -- B.6. Homogeneous tubes -- Bibliography -- Back Cover
N2  - 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
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ER  -