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Quiver Grassmannians of Extended Dynkin Type $D$: Part I: Schubert Systems and Decompositions Into Affine Spaces: 261 (Memoirs of the American Mathematical Society)

Algebra

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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 $M$.

Product Specifications

Format: paperback
ASIN: 1470436477
Category: Books > Subjects > Science, Nature & Maths > Mathematics > Algebra
Domain: Amazon UK
Release Date: 30 October 2019
Listed Since: 23 July 2019

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Quiver Grassmannians of Extended Dynkin Type $D$: Part I: Schubert Systems and Decompositions Into Affine Spaces: 261 (Memoirs of the American Mathematical Society)

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