Picture groups and green sequences from the perspective of Ringel--Hall algebras

Let $k$ be an algebraically closed field and let $\mathrm{\Lambda }$ be a finite-dimensional associative $k$-algebra. We apply Joyce and Brideland’s notion of Ringel–Hall algebra to prove results about picture spaces and picture groups. For instance, we construct a faithful group functor for $\mathrm{\Lambda }$, i.e. a faithful functor from the $\tau$-cluster morphism category of $\mathrm{\Lambda }$ into a groupoid. Consequently, by results of Hanson–Igusa, the picture space of $\mathrm{\Lambda }$ is locally CAT(0) provided that the $\tau$-cluster morphism category of $\mathrm{\Lambda }$ admits compatibility of last factors. We also show that green sequences are in bijection with certain positive expressions in the picture group, which generalizes a result of Igusa–Todorov beyond hereditary $k$-algebras.