Combinatorial Hopf Algebra and structure constants

组合 Hopf 代数和结构常数

• 批准号: 170251-2013
• 负责人: Bergeron, Nantel
• 金额: $ 2.48万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635406
• 关键词: Combinatorial; Hopf; Algebra; structure; constants

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: (A) (with S. Assaf and F. Sotille) Structure constants of the multiplication of Schubert polynomials and the Gromov-Witten invariants.(B) (with C. Benedetti) Structure constants of the multiplication of k-Schur functions (C) (With M. Aguiar, C. Benedetti, P. Diaconis, Nat Thiem, A. Lauve and F. Saliola) Hopf structure on the category of super-representations associated to family of groups.

我们的大部分工作都是在代数组合学中，也就是说，代数和组合学之间的相互作用。我们主要有兴趣使用组合工具解决代数问题。此外，我们认为的问题通常来自其他数学和科学领域，例如几何学，拓扑，数学物理学，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小组开发的组合Hopf代数的理论。在该框架内，我们的大多数研究问题都可以重新铸造。例如，Pieri型公式和各种类型的FLAG歧管的共同体结构被认为是对称多项式模块。我们计划调查与上述有关的几个问题，并考虑由科学的各个领域引起的新问题。在此提案中，我们主要将注意力集中在其中的三个上。这种类型的调查非常适合培训年轻研究人员，因为问题的一部分总是基本而令人兴奋的。我们还应该在这些调查中提及我们对计算机的使用。我们的方法是使用符号操纵语言（SAGE）进行实验，以便获得对研究对象和所涉及机制的全面理解。这个实验过程提出了证明或新猜想的想法。我们的小组一直积极参与Sage/Combinat的发展，Sage/Combinat是一种专门针对此类研究的开源环境。我们计划研究的三个具体问题是：（a）（带有S. assaf和F. sotille）的结构常数，用于舒伯特多项式的乘法和Gromov-Witten的不变式。（b）（b）（带有C. benedetti）结构常数K-Schur函数的乘法（C）（C. aguiar，C。aguiar，p. diole thai ant a.与群体家族相关的超代表类别类别的HOPF结构。