Representations of supergroups. (English) Zbl 0547.22014
Summary: An explicit construction of representations of supergroups is given in terms of direct products of covariant and contravariant fundamental representations. The rules of supersymmetrization are characterized by extended Young supertableaux. This constructive approach leads to explicit transformation properties of higher representations as well as to closed explicit formulas for characters from which other invariants such as dimensions and eigenvalues of all Casimir operators can be calculated. We have applied this approach so far to the supergroups S\(U(\underset \tilde{} N/\underset \tilde{} M)\), OS\(P(\underset \tilde{} N/2\underset \tilde{} M)\), \(P(\underset \tilde{} N)\), for which we have obtained all the representations constructible as direct products of the fundamental (defining) representations. An argument is presented toward the irreducibility of all these representations.
MSC:
22E70 | Applications of Lie groups to the sciences; explicit representations |
81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
Keywords:
explicit construction; fundamental representations; extended Young supertableaux; characters; Casimir operatorsReferences:
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