Construction of graded covariant $GL(m/n)$ modules using tableaux

King, R.C. ; Welsh, T.A.

Journal of Algebraic Combinatorics, Tome 1 (1992) no. 2, pp. 151-170.

Résumé

Summary: Irreducible covariant tensor modules for the Lie supergroups $GL( m/ n)$ and the Lie superalgebras $gl( m/ n)$ and $sl( m/ n)$ are obtained through the use of Young tableaux techniques. The starting point is the graded permutation action, first introduced by Dondi and Jarvis, on $V ^$ otimesl . The isomorphism between this group of actions and the symmetric group S $\_{l}$ enables the graded generalization of the Young symmetrizers, and hence of the column relations and Garnir relations, to be made. Consequently, corresponding to each partition of l an irreducible GL( m/ n) module may be obtained as a submodule of V ^ otimesl . A basis for the module labeled by the partition $lambda$ is provided by $GL( m/ n)$-standard tableaux of shape $lambda$ defined by Berele and Regev. The reduction of an arbitrary tableau to standard form is accomplished through the use of graded column relations and graded Garnir relations. The standardization procedure is algorithmic and allows matrix representations of the Lie superalgebras $gl( m/ n)$ and $sl( m/ n)$ to be constructed explicitly over the field of rational numbers. All the various steps of the standardization algorithm are exemplified, as well as the explicit construction of matrices representing particular elements of $gl( m/ n)$ and $sl( m/ n)$.

Détail

Keywords: Young tableaux, Lie superalgebras, modules