Reduced expressions in a Coxeter system with a complete Coxeter graph

Let (W,S) be a Coxeter system with a complete Coxeter graph. Let Red(z) be the set of all reduced expressions of z∈W. Our aim is to study the structure of Red(z) and to compute the cardinality |Red(z)| of Red(z). We reduce it to the case of z being a bc-element. By associating each ζ′∈Red(z) to a symbol Φ(ζ′), we establish a bijection between the set Red(z) and the equivalence class Symb(Φ(ζ)) of symbols containing Φ(ζ) for any ζ∈Red(z). By this bijection, we reduce the computation of |Red(z)| to that of |Symb(Ψ(ζ))|. Then we deduce some recursive or exact formulae of |Symb(α)| for any symbol α.