Given an irreducible well-generated complex reflection group W with Coxeter number h, we call a Coxeter element any regular element (in the sense of Springer) of order h in W; this is a slight extension of the most common notion of Coxeter element. We show that the class of these Coxeter elements forms a single orbit in W under the action of reflection automorphisms. For Coxeter and Shephard groups, this implies that an element c is a Coxeter element if and only if there exists a simple system S of reflections such that c is the product of the generators in S. We moreover deduce multiple further implications of this property. In particular, we obtain that all noncrossing partition lattices of W associated to different Coxeter elements are isomorphic. We also prove that there is a simply transitive action of the Galois group of the field of definition of W on the set of conjugacy classes of Coxeter elements. Finally, we extend several of these properties to regular elements of arbitrary order.
|Original language||English (US)|
|Number of pages||12|
|Journal||Discrete Mathematics and Theoretical Computer Science|
|State||Published - 2015|
|Event||27th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2015 - Daejeon, Korea, Republic of|
Duration: Jul 6 2015 → Jul 10 2015
Bibliographical noteFunding Information:
†Email: firstname.lastname@example.org. Supported by NSF grant DMS-1001933. ‡Email: email@example.com. Supported by the Austrian Science Foundation FWF, grants Z130-N13 and F50-N15, the latter in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”. §Email: firstname.lastname@example.org. Supported by the German Research Foundation DFG, grant STU 563/2-1 “Coxeter-Catalan combinatorics”.
© 2015 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.
- Coxeter elements
- Coxeter groups
- Noncrossing partitions
- Reflection groups
- Shephard groups