Supersymmetric Schur functions and Lie superalgebra representations

Lie superalgebras and their representations continue to play an important role in the understanding and exploitation of supersymmetry in physical systems. The Lie superalgebras that we consider, namely gl(m|n) and sl(m|n) (sometimes denoted as U(m|n) and SU(m|n)), have applications in quantum mechanics, nuclear physics, string theory, conformal field theory, supergravity, M-theory, lattice QCD, solvable lattice models, spin systems and quantum systems. The representation theory of Lie superalgebras and in particular of gl(m|n) or its simple counterpart sl(m|n), is not a straightforward copy of the corresponding theory of Lie algebras. In the early days of Lie superalgebra representation theory, it was believed that the standard methods of covariant, contravariant and mixed tensor representations with the corresponding Young techniques yield the characters of gl(m|n) irreducible representations in terms of supersymmetric S-functions. Although this is certainly true for the covariant and contravariant tensor representations2, where the supersymmetric S-function is labelled by a single partition λ, it is not so for the mixed tensor representations3 where the corresponding S-function is labelled by a composite partition ¯ν ;μ. In our research we have first concentrated on supersymmetric S-functions labelled by a partition, for which a determinantal formula was constructed4 using a technique of Kac-Wakimoto. Once this determinantal formula for sλ(x/y) was found, we realized that its validity could also be proved in different ways4, for example using the characterization given by Macdonald5. Furthermore, our work led to new dimension and superdimension formulas6. Following this, we showed that there is still another family of typical representations for which the character is given by a (composite) S-function, namely the so-called critical gl(m|n) representations7. It is conjectured that, under certain conditions, the character indeed coincides with a supersymmetric Schur function indexed by a composite partition. Again, this work gives rise to new dimension and superdimension formulas.

Author dc:creator

• Moens, Els

Contributors dc:contributor

• Van Der Jeugt, J

Statement dc:rights

• info:eu-repo/semantics/openAccess

Language dc:language

und

Identifier

https://biblio.ugent.be/publication/469926
http://doi.org/1854/11331
https://biblio.ugent.be/publication/469926/file/1879158

OAI identifier oai:identifier

oai:archive.ugent.be:469926