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

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Abstract

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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Notes

  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.

References

  1. Armstrong, D.: Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Amer. Math. Soc. 202(949) (2009). doi:10.1090/S0065-9266-09-00565-1

  2. Björner, A., Brenti, F.: Combinatorics of Coxeter Groups, Volume 231 of Graduate Texts in Mathematics. Springer, New York (2005)

    MATH  Google Scholar 

  3. Baumeister, B., Dyer, M., Stump, C., Wegener, P.: A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements. Proc. Amer. Math. Soc. Ser. B 1, 149–154 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benson, D.J.: Polynomial Invariants of Finite Groups. Cambridge University Press, Cambridge (1993). (Number 190)

    Book  MATH  Google Scholar 

  5. Bessis, D.: The dual braid monoid. Ann. Sci. École Norm. Sup. (4) 36(5), 647–683 (2003)

    MathSciNet  MATH  Google Scholar 

  6. Bessis, D.: Finite complex reflection arrangements are \(K(\pi,1)\). Ann. of Math. (2) 181(3), 809–904 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Baumeister, B., Gobet, T., Roberts, K., Wegener, P.: On the Hurwitz action in finite Coxeter groups. Preprint, available at arXiv:1512.04764, December (2015)

  8. Bessis, D., Michel, J.: Explicit presentations for exceptional braid groups. Exp. Math. 13(3), 257–266 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Broué, M., Malle, G., Rouquier, R.: On complex reflection groups and their associated braid groups. CMS Conf. Proc. Amer. Math. Soc. 16, 1–13 (1995)

    MathSciNet  MATH  Google Scholar 

  10. Broué, M., Malle, G., Rouquier, R.: Complex reflection groups, braid groups Hecke algebras. J. Reine Angew. Math. 500, 127–190 (1998)

    MathSciNet  MATH  Google Scholar 

  11. Bessis, D., Reiner, V.: Cyclic sieving of noncrossing partitions for complex reflection groups. Ann. Comb. 15(2), 197–222 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brady, T., Watt, C.: \(K(\pi ,1)\)’s for Artin groups of finite type. In: Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), vol. 94, pp. 225–250 (2002)

  13. Chevalley, C.: Invariants of finite groups generated by reflections. Amer. J. Math. 77, 778–782 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  14. Coxeter, H.S.M.: The product of the generators of a finite group generated by reflections. Duke Math. J. 18, 765–782 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  15. Coxeter, H.S.M.: Finite groups generated by unitary reflections. Abh. Math. Sem. Univ. Hamburg 31, 125–135 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  16. Coxeter, H.S.M.: Regular Complex Polytopes, 2nd edn. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  17. Chapuy, G., Stump, C.: Counting factorizations of Coxeter elements into products of reflections. J. Lond. Math. Soc. (2) 90(3), 919–939 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Humphreys, J.E.: Reflection Groups and Coxeter Groups, Volume 29 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  19. Kane, R.: Reflection Groups and Invariant Theory. CMS Books in Mathematics. Springer, Berlin (2001)

    Book  Google Scholar 

  20. Krattenthaler, C., Müller, T.W.: Cyclic Sieving for Generalised Non-crossing Partitions Associated to Complex Reflection Groups of Exceptional Type. Volume W80, in Memory of Herb Wilf, pp. 209–247. Springer, Berlin (2013)

    MATH  Google Scholar 

  21. Koster, D.W.: Complex reflection groups. Ph.D. thesis, University of Wisconsin (1975). MR2625485, available from ProQuest LLC

  22. Lehrer, G.I., Michel, J.: Invariant theory and eigenspaces for unitary reflection groups. C. R. Math. Acad. Sci. Paris 336(10), 795–800 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lehrer, G.I., Springer, T.A.: Reflection subquotients of unitary reflection groups. Canad. J. Math. 51, 1175–1193 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lehrer, G.I., Taylor, D.E.: Unitary Reflection Groups, Volume 20 of Australian Mathematical Society Lecture Series, vol. 20. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  25. Malle, G.: On the rationality and fake degrees of characters of cyclotomic algebras. J. Math. Sci.-Univ. Tokyo 6(4), 647–678 (1999)

    MathSciNet  MATH  Google Scholar 

  26. Michel, J.: Data for reflection groups. (2014). www.math.jussieu.fr/jmichel/papiers/table.pdf

  27. Michel, J.: The development version of the CHEVIE package of GAP3. J. Algeb. 435, 308–336 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Miller, A.R.: Reflection arrangements and ribbon representations. Europ. J. Combin. 39, 24–56 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Malle, G., Michel, J.: Constructing representations of Hecke algebras for complex reflection groups. LMS J. Comput. Math. 13, 426–450 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Marin, I., Michel, J.: Automorphisms of complex reflection groups. Represent. Theory 14, 747–788 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mühlherr, B.: On isomorphisms between Coxeter groups. Des. Codes Cryptogr. 21(1–3), 189–189 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mühle, H.: EL-shellability and noncrossing partitions associated to well-generated complex reflection groups. Europ. J. Combin. 43(C), 249–278 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pilaud, V., Stump, C.: Brick polytopes of spherical subword complexes and generalized associahedra. Adv. Math. 276, 1–61 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Reading, N.: Clusters, Coxeter-sortable elements and noncrossing partitions. Trans. Amer. Math. Soc. 359, 5931–5958 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shephard, G.C.: Regular complex polytopes. Proc. Lond. Math. Soc. 3(2), 82–97 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  36. Springer, T.A.: Regular elements of finite reflection groups. Invent. Math. 25, 159–198 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  37. Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Can. J. Math. 6, 274–304 (1954)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

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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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Victor Reiner

  2. Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Vivien Ripoll

  3. Institut für Mathematik, Freie Universität, Berlin, Germany

    Christian Stump

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  1. Victor Reiner
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  2. Vivien Ripoll
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  3. Christian Stump
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Correspondence to Christian Stump.

Additional information

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). https://doi.org/10.1007/s00209-016-1736-4

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