Quiver Grassmannians of Extended Dynkin type 𝐷 Part 1: Schubert Systems and Decompositions Into Affine Spaces

Abstract

Let Q be a quiver of extended Dynkin type D n . In this first of two papers, we show that the quiver Grassmannian Gr e (M) has a decomposition into affine spaces for every dimension vector e and every indecomposable representation M of defect −1 and defect 0, with exception of the non-Schurian representations in homogeneous tubes. We 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 Gr e (M) and to solve this system of equations successively in linear terms. This leads to an intricate combinatorial problem, for whose solution we develop the theory of Schubert systems.In the sequel [30], we extend the result of this paper to all indecomposable representations M of Q and determine explicit formulae for their F-polynomials. CONTENTS27 4. Schubert decompositions for type D n 30 5. Proof of Theorem 4.4 37 Appendix A. Representations for quivers of type D n 58 Appendix B. Bases for representations of type D n 61 References 72 arXiv:1507.00392v1 [math.RT]

Representation type via Euler characteristics and singularities of quiver Grassmannians

“…Then all indecomposable representations of , but those in the homogeneous tubes, are string modules. For these particular string modules, we can apply the techniques of to establish cell decompositions into affine spaces.…”

“…The authors [19,20] establish cell decompositions into affine spaces for quiver Grassmannians of (extended) Dynkin type D. Such a cell decomposition implies that the cohomology is concentrated in even degrees and therefore the Euler characteristic is non-negative. Recently, Giovanni Cerulli Irelli, Francesco Esposito, Hans Franzen and Markus Reineke have established cell decompositions for type E in [15].…”

“…The authors establish cell decompositions into affine spaces for quiver Grassmannians of (extended) Dynkin type . Such a cell decomposition implies that the cohomology is concentrated in even degrees and therefore the Euler characteristic is non‐negative.…”

Plücker Relations for Quiver Grassmannians

“…Elliptic curve and two projective lines). Let Q be the generalized Kronecker with arrows a, b, c and d. Let M be the representation of dimension vector(3,4) that is given by the following matrices:…”

“…We consider the quiver Grassmannian Gr e (M) for dimension vector e = (1, 3), which embeds into the product GrassmannianGr(1, 3) × Gr(3, 4) = [ ∆ 1 : ∆ 2 : ∆ 3 ∆ 456 : ∆ 457 : ∆ 467 : ∆ 567 ≃ PThere are no classical Plücker relations for Gr(1, 3) × Gr(3,4). The quiver Plücker relations are as follows:E I (a, / 0, {4, 5, 6, 7}) = ∆ 3 ∆ 456 − ∆ 1 ∆ 567 = 0, E I (b, / 0, {4, 5, 6, 7}) = ∆ 3 ∆ 457 − ∆ 2 ∆ 567 = 0, E I (c, / 0, {4, 5, 6, 7}) = ∆ 3 ∆ 467 − ∆ 1 ∆ 456 = 0, E I (d, / 0, {4, 5, 6, 7}) = ∆ 1 ∆ 567 − ∆ 1 ∆ 467 + ∆ 2 ∆ 457 = 0.…”