Construction of Graded Covariant GL(m/n) Modules Using Tableaux
R.C. King
and T.A. Welsh
DOI: 10.1023/A:1022424304176
Abstract
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 
l . 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 l . A basis for the module labeled by the partition 
is provided by GL( m/ n)-standard tableaux of shape 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).
l . 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 l . A basis for the module labeled by the partition is provided by GL( m/ n)-standard tableaux of shape 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).Pages: 151–170
Keywords: Young tableaux; Lie superalgebras; modules
Full Text: PDF
References
1. A.B. Balentekin and I, Bars, "Dimension and character formulae for Lie supergroups," J. Math. Phys. 22 (1981), 1149-1162.
2. I. Bars, B. Morel and H. Ruegg, "Kac-Dynkin diagrams and supertableaux," J. Math. Phys. 24 (1983), 2253-2262.
3. A. Berele, "Construction of Sp-modules by tableaux, "Linear and Multilinear Algebra 19 (1986), 299-307.
4. A. Berele and A. Regev, "Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras," Adv. Math. 64 (1987), 118-175.
5. J.F. Cornwell, Group Theory in Physics, Volume III: Supersymmetries and Infinite-Dimensional Algebras, Techniques of Physics 10, Academic Press, London, 1989.
6. P.H. Dondi and P.D. Jarvis, "Diagram and superfield techniques in the classical superalgebras,"
7. Phys. A 14 (1981), 547-563.
7. J.A. Green, Polynomial Representations of GL(n), Lecture Notes in Mathematics, 830, Springer, Berlin, 1980.
8. G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, MA, 1981.
9. R.C. King and T.A. Welsh, "Construction of GL(n)-modules using composite tableaux," Linear and Multilinear Algebra, to appear.
10. R.C. King and T.A. Welsh, "Construction of orthogonal group modules using tableaux," Linear and Multilinear Algebra, to appear.
2. I. Bars, B. Morel and H. Ruegg, "Kac-Dynkin diagrams and supertableaux," J. Math. Phys. 24 (1983), 2253-2262.
3. A. Berele, "Construction of Sp-modules by tableaux, "Linear and Multilinear Algebra 19 (1986), 299-307.
4. A. Berele and A. Regev, "Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras," Adv. Math. 64 (1987), 118-175.
5. J.F. Cornwell, Group Theory in Physics, Volume III: Supersymmetries and Infinite-Dimensional Algebras, Techniques of Physics 10, Academic Press, London, 1989.
6. P.H. Dondi and P.D. Jarvis, "Diagram and superfield techniques in the classical superalgebras,"
7. Phys. A 14 (1981), 547-563.
7. J.A. Green, Polynomial Representations of GL(n), Lecture Notes in Mathematics, 830, Springer, Berlin, 1980.
8. G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, MA, 1981.
9. R.C. King and T.A. Welsh, "Construction of GL(n)-modules using composite tableaux," Linear and Multilinear Algebra, to appear.
10. R.C. King and T.A. Welsh, "Construction of orthogonal group modules using tableaux," Linear and Multilinear Algebra, to appear.