Symmetric Functions and Macdonald Polynomials

莫尔斯正在进行一个项目，涉及过滤空间的对称函数，取决于一个正整数k。 在早期的工作中，Lapointe、Lascoux和莫尔斯引入了位于这些子空间中的新多项式族，并根据这些元素的特征提出了一些断言。 其中一个假设是它们的元素构成了对称函数空间的k-子空间的基础，并且麦克唐纳多项式在这个新的基础上正扩展。 更一般地说，他们的主张导致了一个家庭的积极性的假设，这将提供一个自然的完善，许多基本概念的对称函数理论。 例如，它们的元素在k-子空间中的作用类似于Schur函数在对称函数空间中的重要作用。 莫尔斯正致力于证明这些命题，也正在研究这些多项式可能的表示论解释。对称函数理论是数学的经典部分，在物理，工程，对称函数研究的最新进展是在1988年引入了一个新的对称函数族，称为Macdonald函数。值得注意的是，这些多项式也被发现在几何学、表象论和多体物理学等领域发挥着重要作用。 这个项目的成功完成将证明麦克唐纳多项式的重要性质，更一般地说，将为对称函数理论中的基本概念提供自然的改进。

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.