Lefschetz invariants and Young characters for representations of the hyperoctahedral groups

The ring R(B-n) of virtual C-characters of the hyperoctahedral group B-n 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(B-n) is a homomorphic image. In particular, the concept of Young characters of B-n arises from a certain set u(n) of subgroups of B-n, and the Z-basis of R(B-n) consisting of Young characters, which is presented by L. Geissinger and D. Kinch [7], forces R(B-n) to be isomorphic to a partial Burnside ring Omega(B-n, U-n). The linear C-characters of B-n are analyzed with reduced Lefschetz invariants which characterize the unit group of Omega(B-n, U-n). The parabolic Burnside ring PB(B-n) is a subring of Omega(B-n, U-n), and the unit group of PB(B-n) is isomorphic to the four group. The unit group of the parabolic Burnside ring of the even-signed permutation group D-n is also isomorphic to the four group. (C) 2018 Elsevier Inc. All rights reserved.