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We introduce the notions of boundary vertex, linear equivalence and effectiveboundary vertex in the context of Viennot's heaps of pieces. We prove that inthe heap of a fully commutative element in a star reducible Coxeter group,every boundary vertex is linearly equivalent to an effective boundary vertex.Using this result, we establish Property W (in the sense of math.QA/0509363)for star reducible Coxeter groups; this corrects a mistake in the latter paper.

Structural properties of acyclic heaps of pieces with Kazhdan--Lusztig theory