Reduced decompositions in Weyl groups

Let R be a root system with fixed basis ~' and let W be its Weyl group. For every element w ~ W, there exists a natural correspondence between reduced decompositions of w and some linear orders of the set of inversions of w. We study combinatorial properties of these orders. Furthermore, in the case of a symmetric group, for any permutation w we construct a linear action of the symmetric group ,Sl(w ) on the space spanned by reduced decompositions of w.