Combinatorial Hopf Algebra and structure constants

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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: