Ronald C. KING: Factorial characters of classical Lie groups

The characters of irreducible representations of the general linear group specified by partitions are nothing other than Schur functions of the eigenvalues of the group elements. Biedenharn and Louck introduced factorial Schur functions, which were generalised somewhat by Macdonald. These serve to define factorial characters of the general linear group. Here we offer definitions of factorial characters of the other classical Lie groups, namely the orthogonal and symplectic groups. It is shown that with these definitions all these factorial characters satisfy what is essentially the same factorial flagged Jacobi-Trudi identity. This is used to provide combinatorial definitions of the factorial characters in terms of suitably weighted tableaux. To achieve these results use will be made of certain truncated generating functions and the evaluation of determinants by means of non-intersecting lattice path methods. At least in the case of the general linear and symplectic groups, corresponding factorial Tokuyama identities will be described.