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Quiver Grassmannians of Extended Dynkin Type Schubert Systems and Decompositions into Affine Spaces.

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical Society SeriesPublisher: Providence : American Mathematical Society, 2019Copyright date: ©2019Edition: 1st edDescription: 1 online resource (90 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470453992
Subject(s): Genre/Form: Additional physical formats: Print version:: Quiver Grassmannians of Extended Dynkin Type LOC classification:
  • QA613.6 .L67 2019
Online resources:
Contents:
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.
Summary: 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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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.

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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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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