Geometric combinatorics of Weyl groupoids
István Heckenberger
and Volkmar Welker
DOI: 10.1007/s10801-010-0264-2
Abstract
We extend properties of the weak order on finite Coxeter groups to Weyl groupoids admitting a finite root system. In particular, we determine the topological structure of intervals with respect to weak order, and show that the set of morphisms with fixed target object forms an ortho-complemented meet semilattice. We define the Coxeter complex of a Weyl groupoid with finite root system and show that it coincides with the triangulation of a sphere cut out by a simplicial hyperplane arrangement. As a consequence, one obtains an algebraic interpretation of many hyperplane arrangements that are not reflection arrangements.
Pages: 115–139
Keywords: keywords Coxeter complex; simplicial arrangements; weak order; Weyl groupoid
Full Text: PDF
References
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2. Björner, A.: Orderings in Coxeter groups. In: Combinatorics and Algebra. Contemp. Math., vol. 34, pp. 175-195. Am. Math. Soc., Providence (1984)
3. Björner, A.: Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings. Adv. Math. 52, 173-212 (1984)
4. Björner, A.: Topological Methods, vol. 2, pp. 1819-1872. North-Holland, Amsterdam (1995)
5. Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Text in Mathematics, vol.
231. Springer, Berlin (2005)
6. Bourbaki, N.: Groupes et algèbres de Lie, ch. 4, 5 et
6. Éléments de mathématique. Hermann, Paris (1968)
7. Cuntz, M., Heckenberger, I.: Finite Weyl groupoids of rank three. Preprint (2009), 31 pp.
8. Cuntz, M., Heckenberger, I.: Weyl groupoids of rank two and continued fractions. Algebra Number Theory 317-340 (2009)
9. Cuntz, M., Heckenberger, I.: Weyl groupoids with at most three objects. J. Pure Appl. Algebra 213, 1112-1128 (2009)
10. Deligne, P.: Les immeubles des groupes de tresses généralises. Invent. Math. 273-302 (1972)
11. Grünbaum, B.: A catalogue of simplicial arrangements in the real projective plane. Ars Math. Contemp. 2, 1-25 (2009)
12. Heckenberger, I., Schneider, H.-J.: Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid. Preprint (2009), 43 pp.
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