Construction of orthogonal group modules using tableaux

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Abstract
The program of constructing irreducible modules of the classical groups via Young tableaux is extended to include the orthogonal groups O(n) and SO(n). As in Weyl's pioneering work on this topic, each irreducible O(n)-module labelled by a partition λ of l is constructed as a submodule of V ⊗l. The construction is based on new sets of O(n) and SO(n) standard Young tableaux, closely related to those introduced by Proctor. The reduction of an arbitrary O(n) Young tableau to standard form is accomplished through the use of column antisymmetrisation and Garnir relations, as for GL(n), together with a trace removal process, analogous to that introduced by Bercle for Sp(n). For SO(n) it is necessary to define the associate of a Young tableau to effect the final step of the reduction process. The standardisation procedure is algorithmic and allows matrix representations of the Lie algebras so(n) to be constructed explicitly over the field of rational numbers. It is proved that the matrix elements are rational numbers having only integral powers of 2 in their denominators. All the various steps of the standardisation algorithm are exemplified, as well as the explicit construction of matrices representing elements of so(n). A link between the trace condition on a tableau and a Garnir relation on its associate is established.

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