[Paper Review] Quiver Grassmannians and their Euler characteristics: Oberwolfach talk, May 2010

Andrei Zelevinsky|arXiv (Cornell University)|Jun 4, 2010

Algebraic structures and combinatorial models6 references3 citations

TL;DR

This paper investigates quiver Grassmannians—projective varieties parametrizing subrepresentations of quiver representations—and computes their Euler characteristics using generating polynomials (F-polynomials). It establishes explicit formulas for Euler characteristics in key cases, such as Kronecker quivers and Dynkin quivers via generalized minors, and identifies conditions under which these Euler characteristics are non-negative, particularly for rigid indecomposable representations in acyclic quivers.

ABSTRACT

This is an extended abstract of my talk at the Oberwolfach Workshop "Interactions between Algebraic Geometry and Noncommutative Algebra" (May 10 - 14, 2010). We present some properties of quiver Grassmannians and examples of explicit computations of their Euler characteristics.

Motivation & Objective

• To study quiver Grassmannians as projective algebraic varieties parametrizing subrepresentations of quiver representations.
• To compute their Euler characteristics, especially in relation to cluster algebra theory.
• To identify conditions under which these Euler characteristics are non-negative, particularly for rigid and indecomposable representations.
• To extend the framework beyond acyclic quivers using quivers with potentials and decorated representations.
• To investigate the role of the invariant E(ℳ) in classifying representations with vanishing Euler characteristic

Proposed method

• Define quiver Grassmannians as the variety of subrepresentations with a fixed dimension vector e in a given quiver representation M.
• Construct the F-polynomial F_M(u_1,…,u_n) as the generating function of Euler characteristics χ(Gr_e(M)) with monomials u_1^{e_1}⋯u_n^{e_n}.
• Use the property F_{M⊕N} = F_M F_N to reduce the study to indecomposable representations.
• For the Kronecker quiver, derive closed-form expressions for χ(Gr_e(M)) in terms of binomial coefficients for preprojective, preinjective, and regular indecomposables.
• For Dynkin quivers, express F_{M(α)} using generalized minors Δ_{γ,γ} on the corresponding semisimple algebraic group G via a Coxeter element and one-parameter subgroups.
• Introduce decorated (Q,S)-representations and the invariant E(ℳ), which is mutation-invariant and vanishes on negative simple representations, to generalize the class of representations of interest.

Experimental results

Research questions

• RQ1 What are the Euler characteristics of quiver Grassmannians for indecomposable quiver representations, and how do they vary with dimension vectors?
• RQ2 Under what conditions are the Euler characteristics of quiver Grassmannians non-negative?
• RQ3 Can the F-polynomials of quiver Grassmannians in Dynkin quivers be expressed uniformly using generalized minors?
• RQ4 How can the class of representations with non-negative Euler characteristics be extended beyond acyclic quivers?
• RQ5 Does the vanishing of the invariant E(ℳ) characterize representations obtainable by mutation from negative simple representations?

Key findings

• For the Kronecker quiver, the Euler characteristic of Gr_e(M^pr(m)) is given by the product of binomial coefficients: binom(m−e₁, e₂−e₁) binom(e₂−1, e₁).
• For the preinjective case, χ(Gr_e(M^inj(m))) = binom(m−e₂, e₁−e₂) binom(e₁−1, e₂), and for regular representations, it is binom(m−e₁, e₂−e₁) binom(e₂, e₁).
• In the Dynkin case, the F-polynomial F_{M(α)} is expressed as a generalized minor Δ_{γ,γ} evaluated on a specific element in the group G, determined by the Coxeter element and the root α.
• For acyclic quivers, if M is rigid (Ext^1(M,M)=0), then all quiver Grassmannians Gr_e(M) are smooth, and χ(Gr_e(M)) ≥ 0 for all e.
• In the generalized Kronecker quiver with four arrows, a counterexample shows that χ(Gr_e(M)) can be negative: for M of dimension vector (3,4) and e=(1,3), the Euler characteristic is −4.
• The invariant E(ℳ) is mutation-invariant and vanishes on negative simple representations; representations with E(ℳ)=0 are conjectured to be obtainable via mutation from negative simples, though this remains open.