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On non-conjugate Coxeter elements in well-generated reflection groups

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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 Springer’s regular elements of arbitrary order.

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  1. In fact, [30, Theorem 1.3] actually implies the stronger property that \(\varGamma _W\) always acts on the set of W-conjugacy classes.

  2. An imprimitive action is one where there is a nontrivial direct sum decomposition \(V=\oplus _{i=1}^t V_i\) respected by W, in the sense that for each \(w \in W\), there is a permutation \(\sigma \) of \(\{1,2,\ldots ,t\}\) for which \(w(V_i) = V_{\sigma (i)}\).

  3. Private communication with C. Stump at the conference Hyperplane Arrangements: combinatorial and geometric aspects of the DFG Priority Programms “Representation Theory” and “Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory”, February 2013 in Bochum, Germany.

  4. One can observe on the classification (see e.g. [25, Theorem 7.1]) that for any well-generated group, the field \({\mathbb {Q}}(\zeta )\) already contains \(K_W\); but we do not need this here.

  5. J. Michel informed us that the following more general statement can be deduced from considerations in [30]: except for straightforward coincidences, any two irreducible complex reflection groups of different Shephard–Todd types are non-isomorphic as abstract groups.


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The authors thank Jean Michel for many helpful discussions during the preparation of this work, as well as Alex Miller, both for helpful conversations and for pointing them to the results in Koster’s thesis [21]. They also thank Vincent Beck, David Bessis, Christian Krattenthaler and Ivan Marin for enlightening discussions on this subject. Finally, they are grateful to Gunter Malle for useful comments on a previous version of this article.

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Correspondence to Christian Stump.

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Victor Reiner was supported by NSF Grant DMS-1001933. Vivien Ripoll was 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”. Christian Stump was supported by the German Research Foundation DFG, Grant STU 563/2-1 “Coxeter-Catalan combinatorics”.

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Reiner, V., Ripoll, V. & Stump, C. On non-conjugate Coxeter elements in well-generated reflection groups. Math. Z. 285, 1041–1062 (2017).

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