[Paper Review] Quiver Grassmannians of type extended Dynkin type D - Part 1: Schubert systems and decompositions into affine spaces

Oliver Lorscheid, Thorsten Weist|arXiv (Cornell University)|Jul 1, 2015

Algebraic structures and combinatorial models23 references6 citations

TL;DR

This paper establishes that quiver Grassmannians for representations of extended Dynkin quivers of type $; D_n$ with defect $-1$ and $0$ decompose into affine spaces, using Schubert systems to solve equations successively. The decomposition is characterized combinatorially via coefficient quivers, providing explicit formulas for Euler characteristics and $F$-polynomials in the sequel, advancing cluster algebra theory for type $D$.

ABSTRACT

Let $Q$ be a quiver of extended Dynkin type $D$. 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 to this paper, we extend the result of this paper to all indecomposable representations $M$ of $Q$ and determine explicit formulae for their $F$-polynomials.

Motivation & Objective

• To establish a decomposition of quiver Grassmannians $ mathrm{Gr}_{\underline{e}}(M)$ into affine spaces for indecomposable representations $M$ of extended Dynkin type $; D_n$ with defect $-1$ and $0$.
• To characterize the affine cells combinatorially using coefficient quivers associated with a fixed ordered basis of $M$.
• To develop the theory of Schubert systems as a tool for solving systems of equations arising from Schubert cell decompositions.
• To lay the foundation for computing Euler characteristics and $F$-polynomials of quiver Grassmannians in the context of cluster algebras.
• To exclude non-Schurian representations in homogeneous tubes from the decomposition result, identifying them as exceptional cases.

Proposed method

• Construct an ordered basis for each indecomposable representation $M$ of defect $-1$ or $0$ to define a coefficient quiver $\Gamma(M,\mathcal{B})$.
• Define Schubert cells within the quiver Grassmannian $\mathrm{Gr}_{\underline{e}}(M)$ using the order on the basis, leading to a cellular decomposition.
• Model the defining equations of Schubert cells as a system of polynomial equations in linear terms.
• Introduce the concept of Schubert systems to systematically solve the resulting equations step by step.
• Use reflection functors at sources to build coefficient quivers for preprojective representations from simple projectives, preserving acyclicity.
• Prove that the coefficient quivers remain acyclic (hence trees) under successive reflections, enabling normalization of structure constants to 1.

Experimental results

Research questions

• RQ1 Under what conditions does the quiver Grassmannian $\mathrm{Gr}_{\underline{e}}(M)$ for a representation $M$ of type $\widetilde{D}_n$ decompose into affine spaces?
• RQ2 How can the affine cells in such a decomposition be explicitly described using combinatorial data from a coefficient quiver?
• RQ3 What is the role of Schubert systems in solving the defining equations of Schubert cells in quiver Grassmannians?
• RQ4 Why do non-Schurian representations in homogeneous tubes fail to admit such a decomposition?
• RQ5 How does the decomposition into affine spaces relate to the computation of Euler characteristics and $F$-polynomials in cluster algebras of type $D$?

Key findings

• The quiver Grassmannian $\mathrm{Gr}_{\underline{e}}(M)$ decomposes into affine spaces for all indecomposable representations $M$ of defect $-1$ and $0$ in $\widetilde{D}_n$, except for non-Schurian representations in homogeneous tubes.
• The affine cells are parametrized by combinatorial data from the coefficient quiver $\Gamma(M,\mathcal{B})$, which is a tree for the chosen ordered basis.
• The decomposition is constructed via Schubert systems, which provide a systematic method to solve the defining equations of Schubert cells in linear terms.
• The coefficient quivers remain acyclic under successive reflections, ensuring that the structure constants can be normalized to 1, simplifying the equations.
• The method applies uniformly to all representations of defect $-1$ and $0$, and the results are extended in the sequel [LW15] to all indecomposable representations.
• The explicit decomposition enables the computation of Euler characteristics and $F$-polynomials for cluster characters in mutation-finite cluster algebras of type $D$.