Bruhat order on classical Weyl groups: minimal chains and covering relation

In this paper we study some aspects of the Bruhat order on classical Weyl groups, obtaining a direct combinatorial description of the minimal chains, that is chains with the lexicographically minimal labelling. Moreover, we find a combinatorial characterization of the covering relation in the hyperoctahedral group and in the even-signed permutation group, providing results analogous to the well-known characterization of the covering relation in the symmetric group.