Nontrivial quiver families and geometric significance

Identify non-trivial families of quivers obtained from partitions of the Remmel–Whitney quiver into arrow classes satisfying the stated splitting and preservation properties, and determine their geometric significance.

Background

The paper observes that the product-rule arguments for the row-rule subquivers do not depend specifically on the particular row-rule partition of the Remmel–Whitney quiver. Instead, analogous rules would hold for any partition of the arrows into classes satisfying suitable splitting and preservation properties: mixed arrows should split into a positive-type arrow followed by a negative-type arrow, and the infusion and diffusion bijections should preserve the relevant arrow classes.

The authors explicitly state that they do not know any non-trivial examples of such quiver families or what geometric meaning those examples might have. Constructing these families and understanding their geometry would broaden the scope of the positivity framework developed in the paper.