Lattices properties of acyclic alternating pipe dreams

This paper proves some conjectures raised by N. Bergeron, N. Cartier, C. Ceballos, and V. Pilaud about acyclic facets of subword complexes of type A Coxeter groups. We use a representation of subword complex facets extending the pipe dreams defined N. Bergeron and S. Billey and prove that the strongly acyclic facets realize a lattice quotient of a weak order interval. We also give a sufficient condition so that all acyclic facets are strongly acyclic.