Picture groups and maximal green sequences

TL;DR

This paper establishes a direct correspondence between picture groups and maximal green sequences in valued Dynkin quivers, extending the relationship to more general hereditary algebras and providing a detailed structure of the associated pictures.

Contribution

It introduces a bijection between maximal green sequences and positive expressions in picture groups, generalizes the result to hereditary algebras, and characterizes pictures as linear combinations of atoms.

Findings

Bijection between maximal green sequences and positive expressions for Coxeter elements.

Extension of the correspondence to hereditary algebras beyond finite type.

Description of pictures as linear combinations of atoms and characterization of atoms.

Abstract

We show that picture groups are directly related to maximal green sequences for valued Dynkin quivers of finite type. Namely, there is a bijection between maximal green sequences and positive expressions (words in the generators without inverses) for the Coxeter element of the picture group. We actually prove the theorem for the more general set up of "vertically and horizontally ordered" sets of positive real Schur roots for any hereditary algebra (not necessarily of finite type). Furthermore, we show that every picture for such a set of positive roots is a linear combination of "atoms" and we give a precise description of atoms as special semi-invariant pictures.