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Connectivity Properties of Factorization Posets in Generated Groups

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Abstract

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.

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Acknowledgements

We thank the anonymous referee for the many comments and suggestions that helped improve both content and exposition of this paper.

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Authors and Affiliations

  1. Institut für Algebra, Technische Universität Dresden, Zellescher Weg 12–14, 01069, Dresden, Germany

    Henri Mühle

  2. Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Vivien Ripoll

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  1. Henri Mühle
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  2. Vivien Ripoll
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Correspondence to Henri Mühle.

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HM is partially supported by a Public Grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” Program (Reference: ANR-10-LABX-0098) and by Digiteo project PAAGT (Nr. 2015-3161D). VR is supported by the Austrian Science Foundation FWF, Grants Z130-N13 and F50-N15, the latter in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”.

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Mühle, H., Ripoll, V. Connectivity Properties of Factorization Posets in Generated Groups. Order 37, 115–149 (2020). https://doi.org/10.1007/s11083-019-09496-1

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