Dual approach to Coxeter and Artin-Tits groups

本课题研究的领域是有限无限群论和(代数)组合学，涉及反射群和Coxeter群，辫子群和Artin-Tits群及其在结合代数和Hecke代数及其表示中的应用，更确切地说，我们旨在系统地探索Coxeter群和Artin-Tits群的对偶方法。我们的目标也是将对偶方法应用到其他领域。对偶松本性质可以用在代数表示论中来理解遗传k-范畴，其中k是一个代数闭域。此外，我们猜想，将关于Mikado辫子的研究推广到无限类型的Artin-Tits群，得到了相应类型的Iwahori-Hecke代数中的Mikado辫子的像具有正性：更准确地说，当用Hecke代数的典范基来表示Mikado辫子的像时。这不仅本身是有意义的，而且对于理解Artin-Tits群的表示理论也是重要的。

This project is in the areas finite and infinite group theory and (algebraic) combinatorics, concerning reflection and Coxeter groups, braid and Artin-Tits groups with applications in associative algebras as well as in Hecke algebras and their representations.More precisely, we aim to explore systematically the dual approach in Coxeter and Artin-Tits groups. Our goal is also to apply the dual approach to other areas.The dual Matsumoto property can be used in representation theory of algebras to understand the hereditary k-categories where k is an algebraically closed field.Further, it is conjectured that the extension of the study of Mikado braids to Artin-Tits groups of infinite types yields that the images of the Mikado braids in the Iwahori-Hecke algebra of the respective type have positivity properties: more precisely, when expressing the image of a Mikado braid in the canonical basis of the Hecke algebra. This beside being of interest in its own right, is also of importance in understanding the representation theory of Artin-Tits groups.