Quiver Grassmannians of Extended Dynkin Type D Part I: Schubert Systems and Decompositions into Affine Spaces

Download or read book Quiver Grassmannians of Extended Dynkin Type D Part I: Schubert Systems and Decompositions into Affine Spaces written by Oliver Lorscheid and published by American Mathematical Soc.. This book was released on 2019-12-02 with total page 78 pages. Available in PDF, EPUB and Kindle. Book excerpt: Let Q be a quiver of extended Dynkin type D˜n. In this first of two papers, the authors show that the quiver Grassmannian Gre–(M) has a decomposition into affine spaces for every dimension vector 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 Gre–(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.