Lefschetz invariants and Young characters for representations of the hyperoctahedral groups

The ring R(Bn) of virtual C-characters of the hyperoctahedral group Bn has two Z-bases consisting of permutation characters, and the ring structure associated with each basis of them defines a partial Burnside ring of which R(Bn) is a homomorphic image. In particular, the concept of Young characters of Bn arises from a certain set Un of subgroups of Bn, and the Z-basis of R(Bn) consisting of Young characters, which is presented by L. Geissinger and D. Kinch [7], forces R(Bn) to be isomorphic to a partial Burnside ring Ω(Bn,Un). The linear C-characters of Bn are analyzed with reduced Lefschetz invariants which characterize the unit group of Ω(Bn,Un). The parabolic Burnside ring PB(Bn) is a subring of Ω(Bn,Un), and the unit group of PB(Bn) is isomorphic to the four group. The unit group of the parabolic Burnside ring of the even-signed permutation group Dn is also isomorphic to the four group.