On Blow-Ups and Injectivity of Quivers

Abstract

This work connects the idea of a "blow-up" of a quiver with that of injectivity, showing that for a class of monic maps Φ, a quiver is Φ-injective if and only if all blow-ups of it are as well. This relationship is then used to characterize all quivers that are injective with respect to the natural embedding of P n into C n .the electronic journal of combinatorics 16 (2009), #R00 Sections and Retractions in QuivRecall from [1, Definition 7.19] that a section in a category is a morphism that is leftinvertible. Also, [1, Definition 7.24] gives the dual notion of a section as a retraction, a the electronic journal of combinatorics 16 (2009), #R00

A characterization of oriented hypergraphic Laplacian and adjacency matrix coefficients

“…The incidence preserving maps introduced in this section are motivated by the directed graphic version of monic, adjacency-preserving, path embeddings into circuit graphs appearing in [9] as a study of Φ-injectivity classes.…”

Generalizing Kirchhoff laws for Signed Graphs

“…If w 1 and w 2 are adjacent, call their edge e. If w 1 and w 2 are not adjacent, regard G as a subgraph G ∪ e w1w2 where edge e w1w2 is added between w 1 and w 2 . This is called the local-loading of G at {w 1 , w 2 }, and is related to the injective loading properties from [11,10]; to simplify notation we will simply write G ∪ e w1w2 with the understanding that e w1w2 may exist in G. Let P(u 1 u 2 , w 1 w 2 ) be the set of u 1 u 2 -paths containing e w1w2 in G ∪ e w1w2 . Lemma 4.1.1.…”

Oriented hypergraphic matrix-tree type theorems and bidirected minors via Boolean order ideals

“…be the oriented hypergraph obtained by adding a bidirected edge to G for every nonadjacent pair of vertices, where σ = σ and ι = ι for all i ∈ I , and σ = 0 for all i ∈ I 0 (see [15] for relationship to the injective envelope). The sign of a (sub-)contributor is the product of the weak walks.…”

“…Section 4 examines the contributors in the adjacency completion of a bidirected graph to obtain a restatement of the All Minors Matrix-tree Theorem in terms of sub-contributors (as opposed to restricted contributors). This implies there is a universal collection of contributors (up to resigning) which determines the minors of all bidirected graphs that have the same injective envelope-see [15] for more on the injective envelope. These sub-contributors determine permanents/determinants of the minors of the original bidirected graph and are activation equivalent to the forest-like objects in [4].…”