组合算法与有效算法：通过信息探索绘制卡茨算法图
Algèbre combinatoire et effective: des graphes aux algèbres de Kac, via l'exploration informatique
摘要：
这份手稿综合了近十五年来在代数组合学的研究，以突出，主题的主题，其观点。在第一部分，我的论文工作的基础上，我使用的工具，从交换代数，特别是从不变量理论，研究同构问题的组合。我首先考虑代数图不变量与乌拉姆的重建猜想，然后，更普遍的是，年龄代数的关系结构。这在置换群的不变理论中引起了结构和算法上的问题。在第二部分，主旨是寻求简单而丰富的组合模型来描述代数结构及其表示。这包括Hecke群代数的Coxeter群，我介绍和有关的仿射Hecke代数，但也有一些有限维Kac代数的关系，包括因素，和理性Steenrod代数。除了是具体的和建设性的，这样的组合模型揭示了某些代数现象，并可以导致优雅和基本的证明。我最喜欢的工具是计算机探索，算法和有效性方面在这篇手稿中发挥了重要作用。特别是，我描述了国际开源项目 *-Combinat，我成立于2000年，其使命是提供一个可扩展的工具箱，计算机探索代数组合，并促进代码共享研究人员在这一领域。我提出了这个项目的开发所带来的具体挑战，以及我被引导开发的原始算法、设计和开发模型解决方案。
英文摘要：
This manuscript synthesizes almost fifteen years of research in algebraic combinatorics, in order to highlight, theme by theme, its perspectives. In part one, building on my thesis work, I use tools from commutative algebra, and in particular from invariant theory, to study isomorphism problems in combinatorics. I first consider algebras of graph invariants in relation with Ulam's reconstruction conjecture, and then, more generally, the age algebras of relational structures. This raises in return structural and algorithmic problems in the invariant theory of permutation groups. In part two, the leitmotiv is the quest for simple yet rich combinatorial models to describe algebraic structures and their representations. This includes the Hecke group algebras of Coxeter groups which I introduced and which relate to the affine Hecke algebras, but also some finite dimensional Kac algebras in relation with inclusions of factors, and the rational Steenrod algebras. Beside being concrete and constructive, such combinatorial models shed light on certain algebraic phenomena and can lead to elegant and elementary proofs. My favorite tool is computer exploration, and the algorithmic and effective aspects play a major role in this manuscript. In particular, I describe the international open source project *-Combinat which I founded back in 2000, and whose mission is to provide an extensible toolbox for computer exploration in algebraic combinatorics and to foster code sharing among researchers in this area. I present specific challenges that the development of this project raised, and the original algorithmic, design, and development model solutions I was led to develop.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
