ABSTRACT
Looking for a common generalization of most of the important families of symmetric functions, Macdonald was led to introduce (in 1988, see [Macdonald 88]) a new family of two-parameter symmetric functions. His original definition made use of a common property of Schur, Zonal, Jack, and Hall-Littlewood symmetric functions, as well as new, more general symmetric functions that had just been considered by Kevin Kadell in his study of q-Selberg integrals. The common feature of all of these linear bases is that they can all be obtained by the same orthogonalization process, for a suitable choice of scalar product. Thus, we are naturally led to consider a more general scalar product that can be specialized to each of these individual cases. One further exciting feature of Macdonald’s proof of the existence and uniqueness of his new symmetric functions was the introduction of self-adjoint operators for which they are common eigenfunctions with distinct eigenvalues. These operators would soon play a crucial role in the solution of the Calogero-Sutherland model of quantum many-body systems in statistical physics, instantly making Macdonald symmetric functions (polynomials) a fundamental part of this theory. It is not surprising that [Macdonald 95, Chapter VI] is the reference of choice here. A general review of recent work can also be found in [Garsia and Remmel 05].
