Young tableaux and modules of groups and lie algebras

Abstract

In this thesis, Young tableaux are used to provide a very convenient explicit descrip- tion of all the irreducible modules of the classical Lie groups and their Lie algebras, and a large class of irreducible modules of the general linear Lie supergroups and their Lie super algebras. An original account of the Specht module techniques for the symmetric groups is also presented. For each irreducible module, a basis is provided by a set of Young tableaux which index the weights of the module. The action of the group or algebra in ques- tion on these 'standard' tableaux is entirely natural. The result is, in general, a linear combination of non-standard tableaux. For each group, a standardisation algorithm is obtained which enables each non-standard tableaux to be expressed in terms of the basis of standard tableaux. For the symmetric groups and the general linear groups, this algorithm is provided by techniques developed by Garnir. This involves the Garnir relations which are closely related to the fundamental Young symmetrisers obtained by Young and based on the Young diagrams. Berele ex- tended this construction by obtaining further relations between the tableaux based on Weyl's removal of trace tensors. These ideas are extended to the mixed tensor representations of the general linear groups and to the orthogonal groups. In this latter case, new sets of standard tableaux are defined. For the spinor modules, it is necessary to develop a further class of relations. For the supergroups, a standardisation technique is obtained by coupling Garnir's methods with a graded symmetric group action. In each construction, the standardisation algorithm involves simple coeffi- cients, often integral. Consequently, the resulting matrix elements are especially simple. Each of the algorithms is exemplified, as well as the explicit construction of matrices representing elements of the various algebras.

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