A hyperoctahedral analogue of the free lie algebra

AbstractUsing the combinatorial techniques developed by Barcelo and Bergeron, we construct a Bn-module, called L(n), related to the Orlik-Solomon algebra of the hyperoctahedral hyperplane complements lattice (os(Bn)) [10, 13]. The Bn-modules L(n) and os(Bn) are analogous to the modules for the symmetric group which occur in the context of the free Lie algebra and the partition lattice. In particular we show that the module L(n) is the transpose of the module os(Bn) tensored by a sign representation. As a by-product we show that the action of Bn on a natural basis of L(n) is block triangular. The blocks are indexed by the conjugacy classes of Bn and have dimension equal to the number of elements in such a class. We also compute the characters of this action restricted to each block