| The free abelian group R(Q) on the set of indecomposable representations of a quiver Q, over a field K, has a ring structure where the multiplication is given by the tensor product. 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 are 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.;We then use rank functions to describe the ring R( Q)red explicitly when Q is a rooted tree quiver (an oriented tree with a unique sink). We prove that in this case, the ring R(Q) red is a finitely generated Z -module (where R(Q) red is the ring R(Q) modulo the ideal of nilpotents). |
