The rank of a quiver representation

TL;DR

This paper introduces a functorial notion of rank for quiver representations, generalizing linear map rank, and explores its properties and applications in understanding the structure of the representation ring.

Contribution

It defines a global rank functor for quiver representations, constructs associated rank functions, and analyzes their algebraic properties and applications.

Findings

Rank functor generalizes linear map rank for quiver representations

Rank functions induce ring homomorphisms from the representation ring to Z

Rank functors commute with Schur operations in characteristic 0

Abstract

We define a functor which gives the "global rank of a quiver representation" and prove that it has nice properties which make it a generalization of the rank of a linear map. We demonstrate how to construct other "rank functors" for a quiver Q, which induce ring homomorphisms (called "rank functions") from the representation ring of Q to Z. These rank functions give discrete numerical invariants of quiver representations, useful for computing tensor product multiplicities of representations and determining some structure of the representation ring. We also show that in characteristic 0, rank functors commute with the Schur operations on quiver representations, and the homomorphisms induced by rank functors are lambda-ring homomorphisms.