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The Weak Order on Weyl Posets

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

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).

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