A hyperoctahedral analogue of the free Lie algebra (Q1180554)

The relation between the free Lie algebra and the hypertetrahedral hyperplane complements lattice is extended to the case of the hyperoctahedral group \(B_n\). Let \(os(B_n)\) denote the Orlik-Solomon algebra of the hyperoctahedral hyperplane complements lattice. The author constructs a \(B_n\)-module \(L(n)\) related to \(os(B_n)\) and shows that \(L(n)\) is the transpose of the module \(os(B_n)\) tensored by a sign representation. It is also shown that the action of \(B_n\) on a natural basis of \(L(n)\) is block triangular, the blocks being indexed by the conjugacy class of \(B_n\), and characters of this action restricted to each block are computed.