Cut-and-Join Operators and Macdonald Polynomials From the 3-Schur Functions

Abstract

Schur polynomials admit a somewhat mysterious deformation to Macdonald and Kerov polynomials, which do not have a direct group theory interpretation but do preserve most of the important properties of Schur functions. Nevertheless, the family of Schur–Macdonald functions is not sufficiently large: for various applications today, we need their not-yet-known analogues labeled by plane partitions, i.e., three-dimensional Young diagrams. Recently, a concrete way to obtain this generalization was proposed, and miraculous coincidences were described, raising hopes that it can lead in the right direction. But even in that case, much work is needed to convert the idea of generalized 3-Schur functions into a justified and effectively working theory. In particular, we can expect that Macdonald functions (and even all Kerov functions, given some luck) enter this theory on an equal footing with ordinary Schur functions. In detail, we describe how this works for Macdonald polynomials when the vector-valued times, which are associated with plane partitions and are arguments of the 3-Schur functions, are projected onto the ordinary scalar times under nonzero angles that depend on the Macdonald parameters q and t. We show that the cut-and-join operators smoothly interpolate between different limit cases. Most of the examples are restricted to level 2.