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

Lattices properties of acyclic alternating pipe dreams