Non-broken circuits of reflection groups and factorization inD n

The set of non-broken circuits of a reflection group W, denoted NBC(W), appears as a basis of the Orlik-Solomon algebra ofW. The factorization of their enumerating polynomial with respect to their cardinality involves the exponentsd _i-1 ofW. A simple explanation of this factorization is known only for the symmetric groupsS _n (Whitney 13]) and for the hyperoctahedral groupsB _n (Lehrer 7]). In this paper, we present an elementary proof of the fact that the set NBC(W) of any refection groupW is in bijection with the group elements ofW. We give a simple characterization of the non-broken circuits of the Weyl groups of typeD _n and we use this characterization to prove the factorization of their enumerating polynomial. Work partially supported by FGIA from ASU. Work partially supported by NSERC (Canada).