THE TENSOR ALGEBRA OF THE IDENTITY REPRESENTATION AS A MODULE OVER THE LIE SUPERALGEBRAS $ \mathfrak{Gl}(n,\,m)$ AND $ Q(n)$

Degenerate affine Hecke–Clifford algebras and type Q Lie superalgebras

“…Since the inductive approach to the study of linear representations of the symmetric group is so effective, it is preferable to study the Sergeev algebra S(d) introduced in [37,25], which provides a natural fix to this problem. As a vector space, S(d) [5,Theorem 3.4].…”

Irreducible tensor representations of general linear Lie superalgebras

“…On the other hand, both E(λ) and E(S λ ) have the same dimension over K. Indeed, we proved earlier in part B of this section that the dimension of E(λ) is equal to the number of tableaux of shape λ with entries in A. Sergeev [10] and Berele and Regev [3] proved independently that E(S λ ) has the same dimension (and a basis indexed by the same set of tableaux). They also proved that E(S λ ) is irreducible (in both cases using the Schur-Weyl duality for the actions of Σ k and gl(E) on k E).…”

“…Tensor representations of the general linear Lie superalgebras play a special role in the theory of representations of these superalgebras because of their relationship with the representation theory of the symmetric groups and use of combinatorial methods. The first construction of irreducible tensor representations was provided independently by Sergeev [10] and by Berele and Regev [3] using the Schur-Weyl duality in a space of tensors of a superspace over a field of characteristic 0. Implicitly, a different construction was contained earlier in a paper by Akin, Buchsbaum and Weyman [1] without mentioning Lie superalgebras.…”

Spin characters of hyperoctahedral wreath products

“…Later Morris also studied double covering groups of Weyl groups following Schur's theory [13]. Sergeev [16] showed that representation theory of the twisted hyperoctahedral group H n is similar to that of the spin group S n (cf. [15,10]), and proved that Schur's Q-functions also served as generating functions for some irreducible spin supercharacters of the hyperoctahedral groups.…”