On the equations and classification of toric quiver varieties

Abstract

Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight there is an associated quasiprojective toric variety together with a canonical embedding into projective space. It is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most 3. In each fixed dimension d up to isomorphism there are only finitely many ddimensional toric quiver varieties. A procedure for their classification is outlined.

Low dimensional flow polytopes and their toric ideals

“…The conjecture was proved in [26] (dealing in fact with a class of transportation polytopes, that includes the Birkhoff polytopes). Using this result, the following was proved in [12] (it is stated there for K = C, but the proof obviously holds for any base field K): We note also that it follows from known general arguments that the toric ideal of any ddimensional flow polytope has a Gröbner basis generated by elements of degree at most d (see Proposition 7.4 for explanation). In view of this and Theorem 1.1, the following questions naturally arise: Question 1.2.…”

Markov degree of configurations defined by fibers of a configuration

“…In view of [8] and [16], Christian Haase (personal communication, 2013) suggested that the Markov degree associated with two-way transportation polytopes and flow polytopes is three. Very recently Domokos and Joó ( [6]) gave a proof of this general conjecture. Adapting the arguments in [16], we give a proof that the Markov degree associated with two-way transportation polytopes is three in Section 4.1.…”

“…Also note that −C(c) and −C(b) may be on the same layer, but in this case the weight of the self-loop aa on the layer is less than or equal to −2 and our proof is not affected. S is not changed, but g B(a) 2 now has B(a) 6 . Then we will decrease S in Case 4-6 below.…”

Quivers and moduli of their thin sincere representations in Macaulay2

“…i1: CG = toricQuiver(completeGraph(4), {1, -2, 3, 0, 0, 0} ); i2: isTight(CG) o1: false The function "makeTight", returns a tight quiver such that the associated flow polytope (to be defined next section) does not change. The tightening process is outlined in [AVS09], see also [DJ16].…”

“…The main objects are acyclic quivers (finite directed graphs) and representations that associate a one-dimensional vector space to each vertex, i.e., a dimension vector equal to all 1's. These representations are called thin sincere ones and their moduli spaces are projective toric varieties associated to the well-studied flow polytopes, see [Hil98], [AH99], [Hil03] and [DJ16]. Additional motivation is given by Craw and Smith [CS08] who showed that every toric variety is the fine moduli space for stable thin representations of an appropriate quiver with relations.…”