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The Weak Order on Weyl Posets

Part of: Lattices Special aspects of infinite or finite groups Algebraic logic

Published online by Cambridge University Press:  29 January 2019

Joël Gay  and
Vincent Pilaud
Joël Gay
Affiliation:
LRI, Univ. Paris-Sud & LIX, École Polytechnique, Palaiseau, France Email: joel.gay@lri.fr
Vincent Pilaud
Affiliation:
CNRS & LIX, École Polytechnique, Palaiseau, France Email: vincent.pilaud@lix.polytechnique.fr
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Abstract

We define a natural lattice structure on all subsets of a finite root system that extends the weak order on the elements of the corresponding Coxeter group. For crystallographic root systems, we show that the subposet of this lattice induced by antisymmetric closed subsets of roots is again a lattice. We then study further subposets of this lattice that naturally correspond to the elements, the intervals, and the faces of the permutahedron and the generalized associahedra of the corresponding Weyl group. These results extend to arbitrary finite crystallographic root systems the recent results of G. Chatel, V. Pilaud, and V. Pons on the weak order on posets and its induced subposets.

Keywords

weak orderfinite Coxeter grouproot system

MSC classification

Primary: 20F55: Reflection and Coxeter groups
Secondary: 03G10: Lattices and related structures 06B99: None of the above, but in this section
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 4 , August 2020 , pp. 867 - 899
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

VP was partially supported by the French ANR grant SC3A (15 CE40 0004 01).

References

Björner, A., Orderings of Coxeter groups. In: Combinatorics and algebra (Boulder, Colo., 1983). Contemp. Math., 34, Amer. Math. Soc., Providence, RI, 1984, pp. 175–195.https://doi.org/10.1090/conm/034/777701Google Scholar
Björner, A. and Brenti, F., Combinatorics of Coxeter groups. Graduate Texts in Mathematics, 231, Springer, New York, 2005.Google Scholar
Bourbaki, N., Groupes et algèbres de Lie, Chapitres IV–VI. Actualités Scientifiques et Industrielles, 1337, Hermann, Paris, 1968.Google Scholar
Chapoton, F., Fomin, S., and Zelevinsky, A., Polytopal realizations of generalized associahedra. Canad. Math. Bull. 45(2002), 537–566. https://doi.org/10.4153/CMB-2002-054-1Google Scholar
Chatel, G., Pilaud, V., and Pons, V., The weak order on integer posets. Algebr. Comb. 2(2019), 1, 1–48.Google Scholar
Dermenjian, A., Hohlweg, C., and Pilaud, V., The facial weak order and its lattice quotients. Trans. Amer. Math. Soc. 370(2018), 1469–1507. https://doi.org/10.1090/tran/7307Google Scholar
Fomin, S. and Zelevinsky, A., Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15(2002), 497–529. https://doi.org/10.1090/S0894-0347-01-00385-XGoogle Scholar
Fomin, S. and Zelevinsky, A., Cluster algebras. II. Finite type classification. Invent. Math. 154(2003), 63–121. https://doi.org/10.1007/s00222-003-0302-yGoogle Scholar
Fomin, S. and Zelevinsky, A., Y-systems and generalized associahedra. Ann. of Math. (2) 158(2003), 977–1018. https://doi.org/10.4007/annals.2003.158.977Google Scholar
Hohlweg, C., Permutahedra and associahedra: generalized associahedra from the geometry of finite reflection groups. In: Associahedra, Tamari lattices and related structures. Prog. Math. Phys., 299, Birkhäuser/Springer, Basel, 2012, pp. 129–159.https://doi.org/10.1007/978-3-0348-0405-9_8Google Scholar
Hohlweg, C. and Lange, C., Realizations of the associahedron and cyclohedron. Discrete Comput. Geom. 37(2007), 517–543.10.1007/s00454-007-1319-6Google Scholar
Hohlweg, C., Lange, C., and Thomas, H., Permutahedra and generalized associahedra. Adv. Math. 226(2011), 608–640. https://doi.org/10.1007/s00454-007-1319-6Google Scholar
Humphreys, J. E., Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, Cambridge, 1990. https://doi.org/10.1017/CBO9780511623646Google Scholar
Krob, D., Latapy, M., Novelli, J.-C., Phan, H.-D., and Schwer, S., Pseudo-permutations I: first combinatorial and lattice properties. In: 13th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2001). 2001.Google Scholar
Loday, J.-L., Realization of the Stasheff polytope. Arch. Math. (Basel) 83(2004), 267–278. https://doi.org/10.1007/s00013-004-1026-yGoogle Scholar
The On-Line Encyclopedia of Integer Sequences. 2010. http://oeis.orgGoogle Scholar
Pilkington, A., Convex geometries on root systems. Comm. Algebra 34(2006), 3183–3202. https://doi.org/10.1080/00927870600778340Google Scholar
Pilaud, V. and Pons, V., Permutrees. Algebraic Combin. 1(2018), 173–224.10.5802/alco.1Google Scholar
Palacios, P. and Ronco, M. O., Weak Bruhat order on the set of faces of the permutohedron and the associahedron. J. Algebra 299(2006), 648–678. https://doi.org/10.1016/j.jalgebra.2005.09.042Google Scholar
Reading, N., Lattice congruences of the weak order. Order 21(2004), 315–344. https://doi.org/10.1007/s11083-005-4803-8Google Scholar
Reading, N., Cambrian lattices. Adv. Math. 205(2006), 313–353. https://doi.org/10.1016/j.aim.2005.07.010Google Scholar
Reading, N., Clusters, Coxeter-sortable elements and noncrossing partitions. Trans. Amer. Math. Soc. 359(2007), 5931–5958. https://doi.org/10.1090/S0002-9947-07-04319-XGoogle Scholar
Reading, N., Sortable elements and Cambrian lattices. Algebra Universalis 56(2007), 411–437. https://doi.org/10.1007/s00012-007-2009-1Google Scholar
Reading, N., From the tamari lattice to Cambrian lattices and beyond. In: Associahedra, Tamari lattices and related structures. Progress in Mathematics, 299, Birkhäuser, 2012, pp. 293–322.https://doi.org/10.1007/978-3-0348-0405-9_15Google Scholar
Reading, N., Finite Coxeter groups and the weak order. In: Lattice theory: special topics and applications. Vol. 2. Birkhäuser/Springer, Cham, 2016, pp. 489–561.10.1007/978-3-319-44236-5_10Google Scholar
Reading, N. and Speyer, D. E., Cambrian fans. J. Eur. Math. Soc. 11(2009), 407–447. https://doi.org/10.4171/JEMS/155Google Scholar
The Sage developers, Sage Mathematics Software, 2016. http://www.sagemath.orgGoogle Scholar