Jacobi-Trudi type formula for a class of irreducible representations of \(\mathfrak{gl}(m\mid n)\). (English) Zbl 1492.17007
The Jacobi-Trudi formula expresses the Schur functions using the elementary symmetric functions or the complete symmetric functions. Alternatively, one can interpret this formula as an expression for the irreducible characters for finite-dimensional representations of the general linear group in terms of the characters of the symmetric tensor representations.
In this paper the author gives a similar expression for a certain class of finite-dimensional irreducible representations of the general linear Lie superalgebra \(\mathfrak{gl}(m|n)\). Namely, it expresses the characters of these representations using the characters of the symmetric powers of the fundamental representation and their duals. The modules considered in this paper correspond to finite-dimensional irreducible representations with highest weight of the form \[ \lambda=(\alpha_1,\alpha_2, \dots, \alpha_m; -k, -k, \dots, -k) \] such that \(0\leq k \leq m\) and \(\alpha_{m-k} \geq 0 \geq \alpha_{m-k+1}\).
The main theorem of this paper is a special case of Theorem 4.3 in [E. M. Moens and J. Van der Jeugt, J. Phys. A, Math. Gen. 37, No. 50, 12019–12039 (2004; Zbl 1077.17010)]. However, according to Lemma 1.14 in [E. M. Moens, Supersymmetric Schur functions and Lie superalgebra representations. Ghent University (Ph.D. Thesis) (2006; doi:1854/11331] there is a gap in the proof. Therefore the more general theorem is still considered a conjecture.
In this paper the author gives a similar expression for a certain class of finite-dimensional irreducible representations of the general linear Lie superalgebra \(\mathfrak{gl}(m|n)\). Namely, it expresses the characters of these representations using the characters of the symmetric powers of the fundamental representation and their duals. The modules considered in this paper correspond to finite-dimensional irreducible representations with highest weight of the form \[ \lambda=(\alpha_1,\alpha_2, \dots, \alpha_m; -k, -k, \dots, -k) \] such that \(0\leq k \leq m\) and \(\alpha_{m-k} \geq 0 \geq \alpha_{m-k+1}\).
The main theorem of this paper is a special case of Theorem 4.3 in [E. M. Moens and J. Van der Jeugt, J. Phys. A, Math. Gen. 37, No. 50, 12019–12039 (2004; Zbl 1077.17010)]. However, according to Lemma 1.14 in [E. M. Moens, Supersymmetric Schur functions and Lie superalgebra representations. Ghent University (Ph.D. Thesis) (2006; doi:1854/11331] there is a gap in the proof. Therefore the more general theorem is still considered a conjecture.
Reviewer: Sigiswald Barbier (Gent)
MSC:
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 17B20 | Simple, semisimple, reductive (super)algebras |
Citations:
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