Artin Groups and Yokonuma–Hecke Algebras. (26th February 2017)

• Record Type: Journal Article
• Authors: Marin, Ivan
• Is Part Of: International mathematics research notices. Volume 2018:Number 13(2018)
• Page Start: 4022
• Page End: 4062
• Publication Date: 2017-02-26
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnx007
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: We attach to every Coxeter system $(W, S)$, an extension $\mathcal{C}_W$ of the corresponding Iwahori–Hecke algebra. We construct a 1-parameter family of (generically surjective) morphisms from the group algebra of the corresponding Artin group onto $\mathcal{C}_W$ . When $W$ is finite, we prove that this algebra is a free module of finite rank which is generically semisimple. When $W$ is the Weyl group of a Chevalley group, $\mathcal{C}_W$ naturally maps to the associated Yokonuma–Hecke algebra. When $W = \mathfrak{S}_n$ this algebra can be identified with a diagram algebra called the algebra of "braids and ties". The image of the usual braid group in this case is investigated. Finally, we generalize our construction to finite complex reflection groups, thus extending the Broué-Malle-Rouquier construction of a generalized Hecke algebra attached to these groups.