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The absolute order on the symmetric group, constructible partially ordered sets and Cohen-Macaulay complexes. (English) Zbl 1189.05173
Summary: The absolute order is a natural partial order on a Coxeter group $$W$$. It can be viewed as an analogue of the weak order on $$W$$ in which the role of the generating set of simple reflections in $$W$$ is played by the set of all reflections in $$W$$. By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric group is homotopy Cohen-Macaulay. This answers in part a question raised by V. Reiner and the first author. The Euler characteristic of the order complex of the proper part of the absolute order on the symmetric group is also computed.

##### MSC:
 05E15 Combinatorial aspects of groups and algebras (MSC2010) 05A18 Partitions of sets 06A07 Combinatorics of partially ordered sets 20F55 Reflection and Coxeter groups (group-theoretic aspects)
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##### References:
 [1] D. Armstrong, Braid groups, clusters and free probability: An outline from the AIM Workshop, January 2005, available at http://www.aimath.org/WWN/braidgroups/ [2] D. Armstrong, Generalized noncrossing partitions and combinatorics of Coxeter groups, preprint, 2007, 156 pp., math.CO/0611106, Mem. Amer. Math. Soc., in press [3] Athanasiadis, C.A.; Brady, T.; Watt, C., Shellability of noncrossing partition lattices, Proc. amer. math. soc., 135, 939-949, (2007) · Zbl 1171.05053 [4] Bessis, D., The dual braid monoid, Ann. sci. ecole norm. sup., 36, 647-683, (2003) · Zbl 1064.20039 [5] Björner, A., Shellable and cohen – macaulay partially ordered sets, Trans. amer. math. soc., 260, 159-183, (1980) · Zbl 0441.06002 [6] Björner, A., Topological methods, (), 1819-1872 · Zbl 0851.52016 [7] Björner, A.; Brenti, F., Combinatorics of Coxeter groups, Grad. texts in math., vol. 231, (2005), Springer-Verlag New York · Zbl 1110.05001 [8] Björner, A.; Wachs, M., On lexicographically shellable posets, Trans. amer. math. soc., 277, 323-341, (1983) · Zbl 0514.05009 [9] Brady, T., A partial order on the symmetric group and new $$K(\pi, 1)$$’s for the braid groups, Adv. math., 161, 20-40, (2001) · Zbl 1011.20040 [10] Brady, T.; Watt, C., $$K(\pi, 1)$$’s for Artin groups of finite type, Proceedings of the conference on geometric and combinatorial group theory, part I, Haifa, 2000, Geom. dedicata, 94, 225-250, (2002) · Zbl 1053.20034 [11] Carter, R.W., Conjugacy classes in the Weyl group, Compos. math., 25, 1-59, (1972) · Zbl 0254.17005 [12] Hochster, M., Rings of invariants of tori, cohen – macaulay rings generated by monomials, and polytopes, Ann. of math., 96, 318-337, (1972) · Zbl 0233.14010 [13] Humphreys, J.E., Reflection groups and Coxeter groups, Cambridge stud. adv. math., vol. 29, (1990), Cambridge University Press Cambridge, England · Zbl 0725.20028 [14] M. Kallipoliti, Doctoral Dissertation, University of Athens, in preparation [15] V. Reiner, Personal communication with the first author, February 2003 [16] Stanley, R.P., Cohen – macaulay rings and constructible polytopes, Bull. amer. math. soc., 81, 133-135, (1975) · Zbl 0304.52005 [17] Stanley, R.P., Combinatorics and commutative algebra, Progr. math., vol. 41, (1996), Birkhäuser Boston · Zbl 0838.13008 [18] Stanley, R.P., Enumerative combinatorics, vol. 1, Cambridge stud. adv. math., vol. 49, (1998), Cambridge University Press Cambridge, second printing · Zbl 1247.05003 [19] Stanley, R.P., Enumerative combinatorics, vol. 2, Cambridge stud. adv. math., vol. 62, (1999), Cambridge University Press Cambridge · Zbl 0928.05001 [20] Wachs, M., Poset topology: tools and applications, (), 497-615 · Zbl 1135.06001
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