# Horn conditions for quiver subrepresentations and the moment map

## TL;DR

This paper establishes inductive conditions to characterize subrepresentations of general quiver representations, generalizing existing criteria and deriving Horn-type inequalities for associated moment cones.

## Contribution

It introduces new inductive conditions for quiver subrepresentations, extending Belkale's and Schofield's results, and refines the understanding of moment cones in this context.

## Key findings

- Provides inductive criteria for quiver subrepresentations
- Generalizes Belkale's intersection criterion for Schubert varieties
- Derives Horn-type inequalities for moment cones

## Abstract

We give inductive conditions that characterize the Schubert positions of subrepresentations of a general quiver representation. Our results generalize Belkale's criterion for the intersection of Schubert varieties in Grassmannians and refine Schofield's characterization of the dimension vectors of general subrepresentations. This implies Horn type inequalities for the moment cone associated to the linear representation of the group $G=\prod_x \operatorname{GL}(n_x)$ associated to a quiver and a dimension vector $\mathbf n=(n_x)$.