[Paper Review] Euler characteristic of quiver Grassmannians and Ringel-Hall algebras of string algebras

Nicolas Poettering|arXiv (Cornell University)|Feb 16, 2010

Algebraic structures and combinatorial models15 references6 citations

TL;DR

This paper computes the Euler characteristics of quiver Grassmannians and flag varieties for tree and band modules over string algebras using torus actions and gradings, proving their positivity. It establishes a combinatorial formula for these invariants and applies the results to explicitly compute products in the Ringel-Hall algebra of string algebras.

ABSTRACT

We compute the Euler characteristics of quiver Grassmannians and quiver flag varieties of tree and band modules and prove their positivity. This generalizes some results by G.C. Irelli [arXiv:0910.2592]. As an application we consider the Ringel-Hall algebra $C(A)$ of some string algebras $A$ and compute in combinatorial terms the products of arbitrary functions in $C(A)$.

Motivation & Objective

• To compute the Euler characteristics of quiver Grassmannians and flag varieties for tree and band modules over quivers.
• To generalize and improve techniques from Irelli (2010) using torus actions and gradings on representations.
• To prove the positivity of Euler characteristics for these varieties in the context of string algebras.
• To apply the results to compute products in the Ringel-Hall algebra of string algebras in combinatorial terms.
• To establish a formula for the Euler characteristic of quiver Grassmannians of band modules via dimension vector decomposition and factorial expressions.

Proposed method

• Uses $\mathbb{C}^*$-actions induced by gradings on representations to reduce Euler characteristic computation to fixed-point subsets.
• Introduces the concept of stable gradings on locally closed subsets of quiver Grassmannians to preserve Euler characteristics.
• Applies the fixed-point formula to tree and band modules via windings of quivers $F: S \to Q$, where $S$ is a tree or $\tilde{A}_{l-1}$-type quiver.
• Derives a combinatorial formula for $\chi_{\mathbf{d}}(F_*(V))$ as a sum over preimages of dimension vectors under $\mathbf{F}$, using $\chi_{\mathbf{t}}(V)$ for $V$ on $S$.
• For band modules, expresses the Euler characteristic using products of factorials and signed factorial terms based on dimension vector differences.
• Extends the method to quiver flag varieties by applying the same torus-fixed point technique to flag varieties of subrepresentations.

Experimental results

Research questions

• RQ1 What is the Euler characteristic of the quiver Grassmannian $\mathop{\rm Gr}_{\mathbf{d}}(M)$ for a tree or band module $M$ over a string algebra?
• RQ2 How can the Euler characteristic of such Grassmannians be computed combinatorially using gradings and torus actions?
• RQ3 What is the structure of the Ringel-Hall algebra of a string algebra, and how can products of functions be computed explicitly?
• RQ4 Under what conditions is the Euler characteristic of a quiver Grassmannian positive?
• RQ5 How do the Euler characteristics of quiver flag varieties relate to those of Grassmannians in the context of tree and band modules?

Key findings

• The Euler characteristic $\chi_{\mathbf{d}}(F_*(V))$ of the quiver Grassmannian of a tree or band module is given by $\sum_{\mathbf{t} \in \mathbf{F}^{-1}(\mathbf{d})} \chi_{\mathbf{t}}(V)$, reducing the problem to computing Euler characteristics on the base quiver.
• For a band module $V$ of type $\tilde{A}_{l-1}$, the Euler characteristic $\chi_{\mathbf{t}}(V)$ is expressed as a product of factorial terms involving dimension vector entries and differences, with $0! = 1$ and $r! = 0$ for negative $r$.
• The formula simplifies to $\frac{1}{|t_i - t_{i+1}|!}$ if $t_{s(s_i)} \leq t_{t(s_i)}$, and 0 otherwise, ensuring positivity when non-zero.
• The Euler characteristic of quiver flag varieties is computed via the same method, with results generalizing Theorem 1.2 to flag varieties.
• The Ringel-Hall algebra $\mathcal{H}(A)$ of a string algebra $A$ admits explicit computation of products of characteristic functions $\Eins_{\mathbf{F},\mathbf{B},\mathbf{n}}$ via combinatorial formulas.
• The product $\Eins_{\mathbf{F},\mathbf{B},\mathbf{n}} \ast \Eins_{\mathbf{F}',\mathbf{B}',\mathbf{n}'}$ is shown to be a linear combination of basis functions in $\mathcal{H}_{\mathbf{d}}(A)$, with independence from parameters like $\lambda \in \mathbb{C}^*$ for band modules.