Symmetric Functions and Macdonald Polynomials

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Morse is working on a project involving a filtration for the space of symmetric functions that depends on a positive integer k. In earlier work, Lapointe, Lascoux, and Morse introduced new families of polynomials that lie in these subspaces,and they made a number of assertions relying on their characterization of these elements. Among these conjectures is one stating that their elements form a basis for the k-subspaces of the symmetric function space and that the Macdonald polynomials expand positively on this new basis. More generally, their claims lead to a family of positivity conjectures which will provide a natural refinement for many fundamental concepts in symmetric function theory. For example, their elements appear to play a role in the k-subspaces analogous to the important role that the Schur functions play for the symmetric function space. Morse is working on proving these conjectures and is also investigating the possible representation theoretic interpretation for these polynomials.The theory of symmetric functions is a classical part of mathematics with a wide variety of applications in fieldsincluding physics, engineering, and computer science.Recent development in the study of symmetric functionswas made with the introduction in 1988 of a new family ofsymmetric functions called the Macdonald polynomials.Remarkably, these polynomials have also been found to play animportant role in areas such as geometry, representation theory, and many-body physics. The successful completion of this project would prove important properties of the Macdonald polynomials and more generally, would provide a natural refinement for the fundamental concepts in symmetric function theory.