G-systems

A G-system is a collection of Z-bases of Zn with some extra axiomatic conditions. There are two kinds of actions “mutations” and “Bongartz co-completions” naturally acting on a G-system, which provide the combinatorial structure of a G-system. It turns out that “Bongartz co-completions” have good compatibility with “mutations”. The constructions of “mutations” are known before in different contexts, including cluster tilting theory, silting theory, τ-tilting theory, cluster algebras and marked surfaces. We found that in addition to “mutations”, there exists another kind of actions “Bongartz co-completions” naturally appearing in these different theories. With the help of actions of “Bongartz co-completions”, some good combinatorial results can be easily obtained. In this paper, we give the constructions of “Bongartz co-completions” in different theories. Then we show that G-systems naturally arise from these theories, and the “mutations” and “Bongartz co-completions” in these theories are compatible with those in G-systems.