# $\mathcal G$-systems

## TL;DR

This paper introduces $\\mathcal{G}$-systems, a combinatorial framework involving mutations and co-Bongartz completions, unifying various theories like cluster algebras and tilting theory.

## Contribution

It constructs co-Bongartz completions across different theories and demonstrates their compatibility with mutations within the $\\mathcal{G}$-system framework.

## Key findings

- $\\mathcal{G}$-systems unify actions in multiple theories.
- Co-Bongartz completions are constructed in various contexts.
- Mutations and co-Bongartz completions are compatible across theories.

## Abstract

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