Symmetric Functions and Macdonald Polynomials

Morse正在致力于一个项目，该项目涉及对依赖于正整数k的对称函数空间的过滤。在早期的工作中，Lapointe、LasCoux和Morse引入了位于这些子空间中的新的多项式族，他们根据这些元素的特征做出了许多断言。在这些猜想中，有一种猜想说它们的元素构成对称函数空间的k-子空间的基，并且麦克唐纳多项式在这个新的基上正向展开。更广泛地说，他们的主张导致了一族正性猜想，这将为对称函数论中的许多基本概念提供自然的提炼。例如，它们的元素似乎在k-子空间中扮演着类似于舒尔函数在对称函数空间中扮演的重要角色。莫尔斯致力于证明这些猜想，并研究这些多项式可能的表示理论解释。对称函数理论是数学的一个经典部分，在物理、工程和计算机科学等领域有着广泛的应用。对称函数研究的最新进展是在1988年引入了一族新的对称函数族，称为麦克唐纳多项式。值得注意的是，这些多项式也被发现在几何、表示理论和多体物理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，更广泛地说，将为对称函数理论中的基本概念提供自然的提炼。

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.