Angela Hicks: Eigenoperators of Macdonald Polynomials

Abstract

There are a number of linear transformations \(\Delta\) defined in the literature by how they act on the modified Macdonald Polynomial basis, which in turn form an eigenbasis for the transformation. In algebraic combinatorics, the Frobenius characteristic of several \(S_n\) modules of historical interest have been expressible as \(\Delta e_n\) for some appropriate choice of \(\Delta\); thus choices of \(\Delta\) where \(\Delta e_n\) is schur positive are thus of particular interest. A famous example is \(\nabla\) occuring in the famous Shuffle Theorem; \(\nabla e_n\) is the Frobenius characteristic of the Diagonal Coinvariants. The so called "Delta Conjecture" of Haglund, Remmel, and Wilson suggests others. We discuss an even larger family of such operators.