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 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.