Dual approach to Coxeter and Artin-Tits groups
Final Report Abstract
In the project we established for spherical Coxeter systems (W, S) and their related Artin groups A(W ) the dual approach for quasi-Coxeter elements. Quasi-Coxeter elements are generalisations of Coxeter elements, which have important properties in common with Coxeter elements. For all the quasi-Coxeter elements w ∈ W in all the irreducible spherical Coxeter systems (W, S) we analysed the intervals [1, w], and defined the related interval groups. We established presentations for the interval groups. The interval groups are always nicely defined on Carter diagrams by adding either cycle commutator relators or twisted cycle commutator relators, depending on whether the quasi-Coxeter element is a Coxeter element or not. Twisted cycle and cycle commutator relators can be written as relations between positive words, which is of importance within Garside theory. For Coxeter elements, where the interval group is the Artin group, some of our group presentations also arise from cluster algebras. For all proper quasi-Coxeter elements, we established that the interval group related to each of them is not isomorphic to the corresponding Artin group. Hence, we obtained a new family of groups, which deserve further study. Our results classify the interval Garside structures one obtains for quasi-Coxeter elements within the dual approach. Along with the description of the presentations of interval groups, we described important properties for quasi-Coxeter elements, their divisors, and their lifts to the interval group as well as for the affine Artin groups of type Ãn.
Publications
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Interval groups related to finite Coxeter groups I. Algebraic Combinatorics, 6(2), 471-506.
Baumeister, Barbara; Neaime, Georges & Rees, Sarah
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Interval groups related to finite Coxeter groups Part II. Transactions of the London Mathematical Society, 10(1), 100-123.
Baumeister, Barbara; Holt, Derek F.; Neaime, Georges & Rees, Sarah
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Isomorphism and non-isomorphism for interval groups of type D. Journal of Algebra, 629, 399–423.
Baumeister, Barbara; Holt, Derek F.; Neaime, Georges & Rees, Sarah
