EL-Labelings and Canonical Spanning Trees for Subword Complexes
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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.
Key wordsSubword complexes Increasing flips Spanning trees EL-labelings Möbius function Enumeration algorithm
Subject Classifications20F55 06A07 05C05 68R05
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 . Finally, we thank the Sage and Sage-Combinat development teams for making available this powerful mathematics software.
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