Relating spin representations of symmetric and hyperoctahedral groups

The aim of this paper is to present recent results of Yamaguchi (A duality of twisted group algebras of symmetric groups and Lie superalgebras, Ph.D. Thesis, University of Tokyo, 1998), on the structure of a certain twisted group algebra of the hyperoctahedral group considered by Sergeev (Math. Sb. 51 (1985) 419–425 (English translation)) in terms of a twisted group algebra of the symmetric group corresponding to a nontrivial cocycle. This result allows one to explain a mysterious dependence of both descriptions of the corresponding characters in the works of Sergeev (Math. Sb. 51 (1985) 419–425 (English translation)) and Schur (J. Reine Angew. Math. 139 (1911) 155–250) on the so-called Schur Q-functions.