[Sveltesse des intervalles bornés d'un demi-treillis cambrien]
Dans cet article, nous donnons une démonstration courte et algébrique du fait que tous les intervalles bornés d'un demi-treillis γ-cambrien sont sveltes pour tout groupe de Coxeter W et tout élément de Coxeter . Cela signifie que, si un tel intervalle a pour longueur k, il existe une chaîne de longueur k consistant en éléments modulaires à gauche, et il y a exactement k éléments sup-irréductibles et k éléments inf-irréductibles. En conséquence, il s'ensuit que chaque intervalle gradué est distributif. Ce problème était ouvert pour tout groupe de Coxeter qui n'est pas un groupe de Weyl.
In this article, we give a short algebraic proof that all closed intervals in a γ-Cambrian semilattice are trim for any Coxeter group W and any Coxeter element . This means that if such an interval has length k, then there exists a maximal chain of length k consisting of left-modular elements, and there are precisely k join- and k meet-irreducible elements in this interval. Consequently, every graded interval in is distributive. This problem was open for any Coxeter group that is not a Weyl group.
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@article{CRMATH_2016__354_2_113_0,
author = {Henri M\"uhle},
title = {Trimness of closed intervals in {Cambrian} semilattices},
journal = {Comptes Rendus. Math\'ematique},
pages = {113--120},
publisher = {Elsevier},
volume = {354},
number = {2},
year = {2016},
doi = {10.1016/j.crma.2015.12.004},
language = {en},
}
Henri Mühle. Trimness of closed intervals in Cambrian semilattices. Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 113-120. doi : 10.1016/j.crma.2015.12.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.12.004/
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☆ This work was funded by the FWF Research Grant No. Z130-N13, and by a Public Grant overseen by the French National Research Agency (ANR) as part of the “Investissements d'Avenir” Program (Reference: ANR-10-LABX-0098).
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