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，代数和组合学之间的相互作用。我们主要对使用组合工具解决代数问题感兴趣。此外，我们考虑的问题通常来自其他数学和科学领域，如几何、拓扑、数学物理、计算机科学等。研究许多代数对象的组合结构的最佳框架之一是我们团队发展的组合Hopf代数理论。我们的大多数研究问题都可以在这个框架内重塑。例如，Pieri型公式和各种类型的旗流形的上同调的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，并考虑不同科学领域出现的新问题。在本提案中，我们主要将注意力集中在其中三个方面。这种类型的调查是培训年轻研究人员的理想选择，因为总有一些问题的部分是基础的，需要研究的。我们还应该提到我们在这些调查中使用计算机。我们的方法是使用符号操纵语言(SAGE)进行实验，以便对研究对象和涉及的机制有一个全面的了解。这一实验过程为证明或新的猜想提供了思路。我们团队一直积极参与SAGE/Combat的开发，这是一个致力于这类研究的开源环境。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: