×

zbMATH — the first resource for mathematics

Connectivity properties of factorization posets in generated groups. (English) Zbl 07204028
Summary: We consider three notions of connectivity and their interactions in partially ordered sets coming from reduced factorizations of an element in a generated group. While one form of connectivity essentially reflects the connectivity of the poset diagram, the other two are a bit more involved: Hurwitz-connectivity has its origins in algebraic geometry, and shellability in topology. We propose a framework to study these connectivity properties in a uniform way. Our main tool is a certain linear order of the generators that is compatible with the chosen element.

MSC:
06A07 Combinatorics of partially ordered sets
20F05 Generators, relations, and presentations of groups
20F36 Braid groups; Artin groups
PDF BibTeX Cite
Full Text: DOI
References:
[1] Armstrong, Drew, Generalized noncrossing partitions and combinatorics of Coxeter groups, Memoirs of the American Mathematical Society, 202, 949, 0-0 (2009) · Zbl 1191.05095
[2] Athanasiadis, CA; Brady, T.; Watt, C., Shellability of noncrossing partition lattices, Proc. Amer. Math. Soc., 135, 939-949 (2007) · Zbl 1171.05053
[3] Baumeister, B.; Gobet, T.; Roberts, K.; Wegener, P., On the Hurwitz action in finite coxeter groups, J. Group Theory, 20, 103-132 (2017) · Zbl 1368.20045
[4] Ben-Itzhak, T.; Teicher, M., Graph theoretic method for determining Hurwitz equivalence in the symmetric group, Israel J. Math., 135, 83-91 (2003) · Zbl 1066.20003
[5] Bessis, D., The dual braid monoid, Annales Scientifiques de l’École Normale Supérieure, 36, 647-683 (2003) · Zbl 1064.20039
[6] Bessis, D., A dual braid Monoid for the free group, J. Algebra, 302, 55-69 (2006) · Zbl 1181.20049
[7] Bessis, D., Finite complex reflection arrangements are K(π, 1), Ann. Math., 181, 809-904 (2015) · Zbl 1372.20036
[8] Biane, P., Some properties of crossings and partitions, Discret. Math., 175, 41-53 (1997) · Zbl 0892.05006
[9] Björner, A., Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc., 260, 159-183 (1980) · Zbl 0441.06002
[10] Björner, A.; Brenti, F., Combinatorics of Coxeter Groups (2005), New York: Springer, New York · Zbl 1110.05001
[11] Björner, A.; Garsia, A.; Stanley, RP, An Introduction to Cohen-Macaulay Posets, Ordered Sets (1982), Dordrecht: Springer, Dordrecht
[12] Björner, A.; Wachs, ML, On lexicographically shellable posets, Trans. Amer. Math. Soc., 277, 323-341 (1983) · Zbl 0514.05009
[13] Björner, A.; Wachs, ML, Shellable and nonpure complexes and posets I, Trans. Amer. Math. Soc., 348, 1299-1327 (1996) · Zbl 0857.05102
[14] Brady, T., A partial order on the symmetric group and new K(π, 1)’s for the braid groups, Adv. Math., 161, 20-40 (2001) · Zbl 1011.20040
[15] Brady, T.; Watt, C., Non-crossing partition lattices in finite real reflection groups, Trans. Amer. Math. Soc., 360, 1983-2005 (2008) · Zbl 1187.20051
[16] Brieskorn, E., Automorphic sets and braids and singularities, Contemp. Math., 78, 45-115 (1988) · Zbl 0716.20017
[17] Deligne, P.: Letter to Eduard Looijenga. Available at http://homepage.rub.de/christian.stump/Deligne_Looijenga_Letter_09-03-1974.pdf (1974)
[18] Hou, d.X.: Hurwitz equivalence in tuples of generalized quaternion groups and dihedral groups. Electron. J. Comb., 15 (2008) · Zbl 1188.20032
[19] Garside, FA, The braid group and other groups, Q. J. Math., 20, 235-254 (1969) · Zbl 0194.03303
[20] Hachimori, M., Decompositions of two-dimensional simplicial complexes, Discret. Math., 308, 2307-2312 (2008) · Zbl 1137.52006
[21] Huang, J.; Lewis, JB; Reiner, V., Absolute order in general linear groups, J. Lond. Math. Soc., 95, 223-247 (2017) · Zbl 06775076
[22] Humphreys, JE, Reflection groups and coxeter groups (1990), Cambridge: Cambridge University Press, Cambridge · Zbl 0725.20028
[23] Hurwitz, A., Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann., 39, 1-60 (1891) · JFM 23.0429.01
[24] Kharlamov, VM; Kulikov, VS, On braid monodromy factorizations, Izvestiya Rossiiskoi Akademii Nauk Seriya Matematicheskaya, 67, 79-118 (2003)
[25] Kreweras, G., Sur les partitions non croisées d’un cycle, Discret. Math., 1, 333-350 (1972) · Zbl 0231.05014
[26] Kulikov, VS, Factorizations in finite groups, Sbornik: Mathematics, 204, 237-263 (2013) · Zbl 1290.14020
[27] Kulikov, VS; Teicher, M., Braid monodromy factorizations and diffeomorphism types, Izvestiya Rossiiskoi Akademii Nauk Seriya Matematicheskaya, 64, 89-120 (2000) · Zbl 1004.14005
[28] Lehrer, GI; Springer, TA, Reflection subquotients of unitary reflection groups, Can. J. Math., 51, 1175-1193 (1999) · Zbl 0958.51017
[29] Lehrer, GI; Taylor, DE, Unitary Reflection Groups (2009), Cambridge: Cambridge University Press, Cambridge
[30] Libgober, A., invariants of plane algebraic curves via representations of the braid groups, Inventiones Mathematicae, 95, 25-30 (1989) · Zbl 0674.14015
[31] Mühle, H., EL-shellability and noncrossing partitions associated with well-generated complex reflection groups, Eur. J. Comb., 43, 249-278 (2015) · Zbl 1301.05367
[32] Mühle, H., Nadeau, P.: A poset structure on the alternating group generated by 3-cycles. arXiv:1803.00540 (2018) · Zbl 07140434
[33] Reiner, V., Non-crossing partitions for classical reflection groups, Discret. Math., 177, 195-222 (1997) · Zbl 0892.06001
[34] Reiner, V.; Ripoll, V.; Stump, C., On non-conjugate coxeter elements in well-generated reflection groups, Mathematische Zeitschrift, 285, 1041-1062 (2017) · Zbl 1377.20027
[35] Sia, C.: Hurwitz equivalence in tuples of dihedral groups, dicyclic groups, and semidihedral groups. Electron. J. Comb., 16 (2009) · Zbl 1191.20035
[36] Stanley, R.P.: Enumerative Combinatorics. 2nd edn., vol. 1. Cambridge University Press, Cambridge (2011) · Zbl 0608.05001
[37] Vince, A.; Wachs, ML, A shellable poset that is not lexicographically shellable, Combinatorica, 5, 257-260 (1985) · Zbl 0623.06003
[38] Walker, JW, A poset which is shellable but not lexicographically shellable, Eur. J. Comb., 6, 287-288 (1985) · Zbl 0579.06001
[39] Wegener, P.: On the Hurwitz action in affine coxeter groups. arXiv:1710.06694 (2017) · Zbl 07173209
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.