arXiv subjects
Publications and source records attributed to Ryan Kinser.
We give a formula for counting tree modules for the quiver S_g with g loops and one vertex in terms of tree modules on its universal cover. This formula, along with work of Helleloid and Rodriguez-Villegas, is used to show that the number of d-dimensional tree modules for S_g is polynomial in g with the same degree and leading coefficient as the counting polynomial A_{S_g}(d, q) for absolutely indecomposables over F_q, evaluated at q=1.
For an acyclic quiver Q, we solve the Clebsch-Gordan problem for the projective representations by computing the multiplicity of a given indecomposable projective in the tensor product of two indecomposable projectives. Motivated by this problem for arbitrary representations, we study idempotents in the representation ring of Q (the free abelian group on the indecomposable representations, with multiplication given by tensor product). We give a general technique for constructing such idempotents and for decomposing the representation ring into a direct product of ideals, utilizing morphisms between quivers and categorical Moebius inversion.
Rank functors on a quiver $Q$ are certain additive functors from the category of representations of $Q$ to the category of finite-dimensional vector spaces. Composing with the dimension function on vector spaces gives a rank function on $Q$. These induce functions on $\rep(Q, α)$, the variety of representations of $Q$ of dimension vector $α$, and thus can be used to define "rank loci" in $\rep(Q, α)$ as collections of points satisfying finite lists of linear inequalities of rank functions. Although quiver rank functions are not generally semicontinuous like the rank of a linear map, we show here that they do have the geometric property that these rank loci are constructible subvarieties. The same is true for loci defined by rank functions in Schofield's subrepresentation bundles on $\rep(Q, α)$, and in quiver Grassmannians.
For each positive integer $n \geq 4$, we give an inequality satisfied by rank functions of arrangements of $n$ subspaces. When $n=4$ we recover Ingleton's inequality; for higher $n$ the inequalities are all new. These inequalities can be thought of as a hierarchy of necessary conditions for a (poly)matroid to be realizable. Some related open questions about the "cone of realizable polymatroids" are also presented.
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.
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 show that if Q is a rooted tree (an oriented tree with a unique sink), then the ring $R(Q)_{red}$ is a finitely generated $\Z$-module (here $R(Q)_{red}$ is the ring R(Q) modulo the ideal of all nilpotent elements). We will describe the ring $R(Q)_{red}$ explicitly, by studying functors from the category Rep(Q) of representations of Q over K to the category of finite dimensional K-vector spaces. We also present an open problem for future direction.