Overview
This study focuses on the topological Euler characteristic of the quiver Grassmannian in the context of generic representations. The research presents a Chern class formula and a finite torus-localization formula applicable to these characteristics. For acyclic quivers, this formula can be combined with existing covariant calculation methods to derive a finite integral linear combination of covariant multiplicities. This is equivalent to determining semi-invariant multiplicities for a single flag-extended quiver.
Research Context
The investigation into the topological Euler characteristic is situated within the broader field of quiver theory. Quiver Grassmannians, which are algebro-geometric objects, are central to this work. The concept of generic representations suggests an analysis applicable to a broad class of instances rather than highly specific or degenerate cases. The established covariant calculation methods by Derksen, Schofield, and Weyman provide a foundation for integrating the new formulas, particularly for acyclic quivers. The concept of semi-invariant multiplicities relates to invariants under group actions, a key aspect in representation theory.
Approach
The research develops two primary formulas related to the topological Euler characteristic of the quiver Grassmannian for generic representations: a Chern class formula and a finite torus-localization formula. The torus-localization approach is a technique in algebraic topology that relates global topological invariants to local data at fixed points of a torus action. For the specific case where the quiver $Q$ is acyclic, the developed formula integrates with the covariant calculation framework previously introduced by Derksen, Schofield, and Weyman. This integration allows for the expression of the Euler characteristic as a finite integral linear combination of covariant multiplicities. This combination is noted to be equivalent to semi-invariant multiplicities for a single flag-extended quiver of $Q$. The methodology also extends to recording the same construction for multiplicative characteristic genera. The study includes the presentation of explicit formulas and examples, specifically for generalized Kronecker quivers.
Findings
- A Chern class formula for the topological Euler characteristic of the quiver Grassmannian for a generic representation was developed.
- A finite torus-localization formula for the topological Euler characteristic of the quiver Grassmannian for a generic representation was established.
- For acyclic quivers, the developed formula integrates with the covariant calculations of Derksen–Schofield–Weyman. This combination results in a finite integral linear combination of covariant multiplicities.
- This finite integral linear combination of covariant multiplicities is equivalent to semi-invariant multiplicities for a single flag-extended quiver of $Q$.
- Explicit formulas and examples were provided, specifically illustrated for generalized Kronecker quivers.
- The same construction was recorded for multiplicative characteristic genera.