Abstract
We describe edge labelings of the increasing flip graph of a subword complex on a finite Coxeter group, and study applications thereof. On the one hand, we show that they provide canonical spanning trees of the facet-ridge graph of the subword complex, describe inductively these trees, and present their close relations to greedy facets. Searching these trees yields an efficient algorithm to generate all facets of the subword complex, which extends the greedy flip algorithm for pointed pseudotriangulations. On the other hand, when the increasing flip graph is a Hasse diagram, we show that the edge labeling is indeed an EL-labeling and derive further combinatorial properties of paths in the increasing flip graph. These results apply in particular to Cambrian lattices, in which case a similar EL-labeling was recently studied by M. Kallipoliti and H. Mühle.
V. Pilaud was supported by the spanish MICINN grant MTM2011-22792, by the french ANR grant EGOS 12 JS02 002 01, by the European Research Project ExploreMaps (ERC StG 208471), and by a postdoctoral grant of the Fields Institute of Toronto..
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Notes
- 1.
We thank an anonymous referee for raising this question.
- 2.
As observed by M. Pocchiola, searching on the positive sink tree or on the negative source tree improves the working space of the algorithm. This issue is relevant for the enumeration of pseudotriangulations and will be discussed in a forthcoming paper of his.
- 3.
The ongoing work on this patch can be found at http://trac.sagemath.org/sage_trac/ticket/11010.
- 4.
The contrary was stated in a previous version of this paper. We thank an anonymous referee for pointing out this mistake.
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Acknowledgements
We are very grateful to the two anonymous referees for their detailed reading of several versions of the manuscript, and for many valuable comments and suggestions, both on the content and on the presentation. Their suggestions led us to the current version of Proposition 15, to correct a serious mistake in a previous version, and to improve several arguments in various proofs.
V. Pilaud thanks M. Pocchiola for introducing him to the greedy flip algorithm on pseudotriangulations and for uncountable inspiring discussions on the subject. We thank M. Kallipoliti and H. Mühle for mentioning our construction in [6]. Finally, we thank the Sage and Sage-Combinat development teams for making available this powerful mathematics software.
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Pilaud, V., Stump, C. (2013). EL-Labelings and Canonical Spanning Trees for Subword Complexes. In: Bezdek, K., Deza, A., Ye, Y. (eds) Discrete Geometry and Optimization. Fields Institute Communications, vol 69. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00200-2_13
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