Combinatorial Hopf Algebra and structure constants

我们的大部分工作是在代数组合学，也就是说，代数和组合之间的相互作用。我们主要对使用组合工具解决代数问题感兴趣。此外，我们考虑的问题往往来自其他领域的数学和科学，如几何，拓扑，数学物理，计算机科学等最好的框架之一，以调查的组合结构的许多代数对象是理论的组合Hopf代数开发我们的小组。我们的大多数研究问题都可以在这个框架内重新设计。例如，Pieri型公式和各种类型的旗流形的上同调结构被认为是对称多项式模。我们计划调查与上述有关的几个问题，并考虑从各个科学领域产生的新问题。在本建议中，我们主要关注其中三个。这种类型的调查是理想的培训年轻的研究人员，因为总有一些问题的部分是基本的和刺激的工作，我们也应该提到我们使用的计算机在这些调查。我们的方法是使用符号操作语言（SAGE）进行实验，以获得对研究对象和所涉及机制的全面理解。这个实验过程为证明或新的理论提供了思路。我们的团队一直积极参与SAGE/combinat的开发，这是一个致力于此类研究的开源环境。我们计划调查的三个具体问题是：

Most of our work is in Algebraic Combinatorics, that is to say, the interplay between algebra and combinatorics. We are primarily interested in solving algebraic problems using combinatorial tools. Moreover, the problems we consider often come from other areas of mathematics and science such as geometry, topology, mathematical physics, computer science, etc. One of the best frameworks to investigate the combinatorial structure of many algebraic objects is the theory of combinatorial Hopf algebras developed by our group. Most of our research problems can be recast within that framework. For example, Pieri-type formulas and the structure of the cohomology of various types of flag manifolds considered as symmetric polynomial modules. We plan to investigate several problems linked to the above and to consider new problems arising from various areas of science. In this proposal we mostly focus our attention on three of them. This type of investigation is ideal for training young researchers, since there are always parts of the problem that are elementary and stimulating to work on. We should also mention our use of computers in these investigations. Our approach is to experiment using a symbolic manipulation language (SAGE) in order to obtain a comprehensive understanding of the objects of study and the mechanisms involved. This experimental process suggests ideas for proofs or new conjectures. Our group has been actively involved in the development of SAGE/combinat, an open source environment dedicated to this type of research. The three specific problems we plan to investigate are: