Jacobi-Trudi type formula for character of irreducible representations of \(\mathfrak{gl}(m|1)\). (English) Zbl 1473.17021
The classical Jacobi-Trudi formula computes Schur symmetric functions in terms of the elementary (resp. complete) symmetric functions. Since these symmetric functions can be realized as irreducible characters of general linear Lie algebras,
In this paper, the authors extend this famous formula to the case of the general linear Lie superalgebras. According to V. Kac, irreducible (finite dimensional) representations of the general linear Lie superalgebra \(gl(m|n)\) are determined by means of dominant weights. They prove a determinantal type formula to compute the irreducible characters of the general Lie superalgebra \(gl(m|1)\) in terms of the characters of the symmetric powers of the fundamental representation and their duals. This formula was conjectured by J. van der Jeugt and E. Moens for the Lie superalgebra \(gl(m|n)\) and generalizes the well-known Jacobi-Trudi formula.
In this paper, the authors extend this famous formula to the case of the general linear Lie superalgebras. According to V. Kac, irreducible (finite dimensional) representations of the general linear Lie superalgebra \(gl(m|n)\) are determined by means of dominant weights. They prove a determinantal type formula to compute the irreducible characters of the general Lie superalgebra \(gl(m|1)\) in terms of the characters of the symmetric powers of the fundamental representation and their duals. This formula was conjectured by J. van der Jeugt and E. Moens for the Lie superalgebra \(gl(m|n)\) and generalizes the well-known Jacobi-Trudi formula.
Reviewer: Shuangjian Guo (Guiyang)
MSC:
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 17B15 | Representations of Lie algebras and Lie superalgebras, analytic theory |
| 17B20 | Simple, semisimple, reductive (super)algebras |
| 17B22 | Root systems |
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