A construction of simple-minded systems over domestic Brauer graph algebras I: the 2-domestic case

Let $A$ be a 2-domestic Brauer graph algebra. We present a construction for a family of objects on $A$-$\stmod$ to be a simple-minded system and our construction provides all simple-minded systems on $A$-$\stmod$. As a byproduct, we provide a new proof of AR-conjecture for 2-domestic Brauer graph algebras and we prove that a weakly simple-minded system with a finite cardinality is a simple-minded system on $A$-$\stmod$. We also prove that an orthogonal system $\mathcal{S}$ which contains at least one object for an Euclidean component extends to a simple-minded system on $A$-$\stmod$ and its extension closure $\mathcal{F}(\mathcal{S})$ is functorially finite.