Horn inequalities and quivers

TL;DR

This paper characterizes the inequalities defining the Mumford cone for quiver representations, extending Horn inequalities to a broader algebraic setting using filtered dimension vectors and inductive methods.

Contribution

It provides a new description of the Mumford cone inequalities for quiver representations, adapting Schofield's argument to the filtered context and offering a new proof of Horn inequalities.

Findings

Derived explicit inequalities for the Mumford cone in quiver representations.

Extended Horn inequalities to the setting of quivers and filtered dimension vectors.

Provided an inductive method for determining subrepresentation dimension vectors.

Abstract

Let G be a complex reductive group acting on a finite-dimensional complex vector space H. Let B be a Borel subgroup of G and let T be the associated torus. The Mumford cone is the polyhedral cone generated by the T-weights of the polynomial functions on H which are semi-invariant under the Borel subgroup. In this article, we determine the inequalities of the Mumford cone in the case of the linear representation associated to a quiver and a dimension vector n=(n_x). We give these inequalities in terms of filtered dimension vectors, and we directly adapt Schofield's argument to inductively determine the dimension vectors of general subrepresentations in the filtered context. In particular, this gives one further proof of the Horn inequalities for tensor products.