Polygon Posets and the Weak Order of Coxeter Groups
Kimmo Eriksson
DOI: 10.1023/A:1022428111696
Abstract
We explore the connection between polygon posets, which is a class of ranked posets with an edge-labeling which satisfies certain 
polygon properties 
, and the weak order of Coxeter groups. We show that every polygon poset is isomorphic to a join ideal in the weak order, and for Coxeter groups where no pair of generators have infinite order the converse is also true.
polygon properties , and the weak order of Coxeter groups. We show that every polygon poset is isomorphic to a join ideal in the weak order, and for Coxeter groups where no pair of generators have infinite order the converse is also true. The class of polygon posets is seen to include the class of generalized quotients defined by Björner and Wachs, while itself being included in the class of alternative generalized quotients also considered by these authors. By studying polygon posets we are then able to answer an open question about common properties of these two classes.
Pages: 233–252
Keywords: polygon poset; weak order; Coxeter group; generalized quotient; finite state automaton
Full Text: PDF
References
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2. A. Bjorner and M.L. Wachs, "Generalized quotients in Coxeter groups," Trans, Amer. Math. Soc. 308 (1988), 1-37.
3. A. Bjorner and G.M. Ziegler, "Introduction to greedoids," in N. White, editor, Matwid Applications, pp. 284-357, Cambridge Univ. Press,
1991. ERIKSSON
4. N. Bourbaki, Groupes etAlgebres de Lie, Hermann, Paris, 1968.
5. B. Brink and R. Hewlett, "A finiteness property and an automatic structure for Coxeter groups," Math. Ann. 296 (1993), 179-190.
6. M. Davis and M. Shapiro, "Coxeter groups are automatic," Ohio State University, preprint, 1991.
7. H. Eriksson, "Computational and combinatorial aspects of Coxeter groups," Ph.D. thesis, KTH, Stockholm, 1994.
8. K. Eriksson, "Convergence of Mozes's game of numbers," Linear Alg. Appl. 166 (1992) 151-165.
9. K. Eriksson, "The numbers game and Coxeter groups," Conference in Algebraic Combinatorics 92, University du Quebec a Montreal,
1992. To appear in Discrete Math.
10. K. Eriksson, "Strongly convergent games and Coxeter groups," PhD thesis, KTH, Stockholm, 1993.
11. P. Headley, PhD thesis to appear, Univ. of Michigan, Ann Arbor, 1994.
12. S. Mozes, "Reflection processes on graphs and Weyl groups," J. Comb. Theory, Series A S3 (1990), 128-142.
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