The absolute order on the hyperoctahedral group
Myrto Kallipoliti
DOI: 10.1007/s10801-010-0267-z
Abstract
The absolute order on the hyperoctahedral group B n is investigated. Using a poset fiber theorem, it is proved that the order ideal of this poset generated by the Coxeter elements is homotopy Cohen-Macaulay. This method results in a new proof of Cohen-Macaulayness of the absolute order on the symmetric group. Moreover, it is shown that every closed interval in the absolute order on B n is shellable and an example of a non-Cohen-Macaulay interval in the absolute order on D 4 is given. Finally, the closed intervals in the absolute order on B n and D n which are lattices are characterized and some of their important enumerative invariants are computed.
Pages: 183–211
Keywords: keywords hyperoctahedral group; absolute order; Cohen-Macaulay poset
Full Text: PDF
References
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4. Athanasiadis, C.A., Brady, T., Watt, C.: Shellability of noncrossing partition lattices. Proc. Am. Math. Soc. 135, 939-949 (2007)
5. Bessis, B.: The dual braid monoid. Ann. Sci. Ec. Norm. Super. 36, 647-683 (2003)
6. Biane, P.: Parking functions of types A and B. Electron. J. Comb. 9, 7 (2002). 5pp (electronic)
7. Björner, A.: Shellable and Cohen-Macaulay partially ordered sets. Trans. Am. Math. Soc. 260, 159- 183 (1980)
8. Björner, A.: Orderings of Coxeter groups. In: Greene, C. (ed.) Combinatorics and Algebra, Boulder
1983. Contemp. Math., vol. 34, pp. 175-195. Am. Math. Soc., Providence (1984)
9. Björner, A.: Topological Methods. In: Graham, R.L., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, pp. 1819-1872. North Holland, Amsterdam (1995)
10. Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol.
231. Springer, New York (2005)
11. Björner, A., Wachs, M., Welker, V.: Poset fiber theorems. Trans. Am. Math. Soc. 357, 1877-1899 (2005)
12. Björner, A., Wachs, M., Welker, V.: On sequentially Cohen-Macaulay complexes and posets. Isr. J. Math. 169, 295-316 (2009)
13. Brady, T.: A partial order on the symmetric group and new K(π, 1)'s for the brad groups. Adv. Math. 161, 20-40 (2001)
14. Brady, T., Watt, C.: K(π, 1)'s for Artin groups of finite type. In: Proceedings of the Conference on Geometric and Combinatorial group theory, Part I (Haifa 2000). Geom. Dedicata, vol. 94, pp. 225-250 (2002)
15. Brady, T., Watt, C.: Non-crossing partition lattices in finite real reflection groups. Trans. Am. Math. Soc. 360, 1983-2005 (2008)
16. Diaconis, P., Graham, R.L.: Spearman's footrule as a measure of disarray. J. R. Stat. Soc. Ser. B 39, 262-268 (1977)
17. Edelman, P.H.: Zeta polynomials and the Möbius function. Eur. J. Comb. 1, 335-340 (1980)
18. Goulden, I.P., Nica, A., Oancea, I.: Enumerative properties of NCB (p, q). Ann. Comb. (2010, to appear). Preprint.
19. Humphreys, J.E.: Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics, vol.
29. Cambridge University Press, Cambridge (1990)
20. Kallipoliti, M.: The absolute order on the hyperoctahedral group (extended abstract). In: Proceedings of FPSAC
2009. Discr. Math. Theoret. Comput. Science, pp. 503-514 (2009)
21. Kallipoliti, M.: Combinatorics and topology of the absolute order on a finite Coxeter group (in Greek). Doctoral Dissertation, University of Athens (2010)
22. Krattenthaler, C.: Non-crossing partitions on an annulus. In preparation
23. Krattenthaler, C., Müller, T.W.: Decomposition numbers for finite Coxeter groups and generalized non-crossing partitions,
2007. Trans. Am. Math. Soc. 362, 2723-2787 (2010)
24. Nica, A., Oancea, I.: Posets of annular noncrossing partitions of types B and D. Discrete Math. 309, 1443-1466 (2009)
25. Quillen, D.: Homotopy properties of the poset of non-trivial p-subgroups of a group. Adv. Math. 28, 101-128 (1978)
26. Reiner, V.: Non-crossing partitions for classical reflection groups. Discrete Math. 177, 195-222 (1997)
27. Stanley, R.P.: Enumerative Combinatorics, vol.
1. Wadsworth & Brooks/Cole, Pacific Grove (1986).
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