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Connectivity properties of factorization posets in generated groups. (English) Zbl 07054649
Summary: We consider three notions of connectivity and their interactions in partially ordered sets coming from reduced factorizations of elements in generated groups. 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 total order of the generators that is compatible with the chosen element.

MSC:
06A06 Partial orders, general
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[1] C.A. Athanasiadis, T. Brady, and C. Watt. “Shellability of Noncrossing Partition Lattices”. Proc. Amer. Math. Soc. 135.4 (2007), pp. 939–949. DOI:10.1090/S0002-9939-06-08534-0. · Zbl 1171.05053
[2] D. Bessis. “Finite Complex Reflection Arrangements are K(π, 1)”. Ann. of. Math. (2) 181.3 (2015), pp. 809–904. DOI:10.4007/annals.2015.181.3.1. · Zbl 1372.20036
[3] A. Björner. “Shellable and Cohen-Macaulay Partially Ordered Sets”. Trans. Amer. Math. Soc. 260.1 (1980), pp. 159–183. DOI:10.2307/1999881.
[4] A. Björner and M.L. Wachs. “On Lexicographically Shellable Posets”. Trans. Amer. Math. Soc. 277.1 (1983), pp. 323–341. DOI:10.2307/1999359.
[5] A. Björner and M.L. Wachs. “Shellable and Nonpure Complexes and Posets I”. Trans. Amer. Math. Soc. 348.4 (1996), pp. 1299–1327. DOI:10.1090/S0002-9947-96-01534-6. · Zbl 0857.05102
[6] T. Brady. “A Partial Order on the Symmetric Group and new K(π, 1)’s for the Braid Groups”. Adv. Math. 161.1 (2001), pp. 20–40. DOI:10.1006/aima.2001.1986. · Zbl 1011.20040
[7] P. Deligne. “Letter to E. Looijenga”. Available athttp : / / homepage . univie . ac . at / christian.stump/Deligne_Looijenga_Letter_09-03-1974.pdf. 1974.
[8] H. Mühle. “EL-Shellability and Noncrossing Partitions Associated with Well-Generated Complex Reflection Groups”. European J. Combin. 43 (2015), pp. 249–278.URL. · Zbl 1301.05367
[9] H. Mühle and V. Ripoll. “Connectivity Properties of Factorization Posets in Generated Groups”. 2017. arXiv:1710.02063.
[10] V. Reiner, V. Ripoll, and C. Stump. “On Non-Conjugate Coxeter Elements in Well-Generated Reflection Groups”. Math. Z. 285.3-4 (2017), pp. 1041–1062. DOI:10.1007/s00209-016-17364. · Zbl 1377.20027
[11] A. Vince and M.L. Wachs. “A Shellable Poset that is not Lexicographically Shellable”. Combinatorica 5.3 (1985), pp. 257–260. DOI:10.1007/BF02579370. · Zbl 0623.06003
[12] J. W. Walker. “A Poset which is Shellable but not Lexicographically Shellable”. European J. Combin. 6.3 (1985), pp. 287–288. DOI:10.1016/S0195-6698(85)80040-8. · Zbl 0579.06001
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