ABSTRACT
In this chapter, we depart from our previous considerations in that we are now going to systematically consider the diagonal action (see Section 8.7) of the symmetric group Sn on the ring R = C[x,y] of polynomials in two sets of variables x and y. This is not an action that corresponds to a reflection group action of Sn, so none of the results considered earlier can be directly applied. It develops that this new context is entirely different, both in the kinds of results that can be obtained and in the difficulty of proving them. It has been the object of intensive study in the last 15 years, and many new avenues of research have opened up as a result. For a more in-depth discussion see M. Haiman’s very nice survey in [Haiman 01].
