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EL-Labelings and Canonical Spanning Trees for Subword Complexes

Chapter
Part of the Fields Institute Communications book series (FIC, volume 69)

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.

Key words

Subword complexes Increasing flips Spanning trees EL-labelings Möbius function Enumeration algorithm 

Subject Classifications

20F55 06A07 05C05 68R05 

Notes

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.

References

  1. 1.
    Björner, A.: Shellable and Cohen-Macaulay partially ordered sets. Trans. Am. Math. Soc. 260(1), 159–183 (1980)zbMATHCrossRefGoogle Scholar
  2. 2.
    Björner, A., Wachs, M.L.: Shellable nonpure complexes and posets. I. Trans. Am. Math. Soc. 348(4), 1299–1327 (1996)zbMATHCrossRefGoogle Scholar
  3. 3.
    Brönnimann, H., Kettner, L., Pocchiola, M., Snoeyink, J.: Counting and enumerating pointed pseudotriangulations with the greedy flip algorithm. SIAM J. Comput. 36(3), 721–739 (electronic) (2006).Google Scholar
  4. 4.
    Ceballos, C., Labbé, J.-P., Stump, C.: Subword complexes, cluster complexes, and generalized multi-associahedra. J. Algebraic Combin. 1–35 (2013). DOI 10.1007/s10801-013-0437-xGoogle Scholar
  5. 5.
    Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)Google Scholar
  6. 6.
    Kallipoliti, M., Mühle, H.: On the topology of the Cambrian semilattices (2012, Preprint). arXiv:1206.6248 Google Scholar
  7. 7.
    Knutson, A., Miller, E.: Subword complexes in Coxeter groups. Adv. Math. 184(1), 161–176 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Knutson, A., Miller, E.: Gröbner geometry of Schubert polynomials. Ann. Math. (2), 161(3), 1245–1318 (2005)Google Scholar
  9. 9.
    Papi, P.: A characterization of a special ordering in a root system. Proc. Am. Math. Soc. 120(3), 661–665 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Pilaud, V., Pocchiola, M.: Multitriangulations, pseudotriangulations and primitive sorting networks. Discret. Comput. Geom. 48(1), 142–191 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Pilaud, V., Santos, F.: Multitriangulations as complexes of star polygons. Discret. Comput. Geom. 41(2), 284–317 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Pilaud, V., Stump, C.: Brick polytopes of spherical subword complexes: a new approach to generalized associahedra (2011, Preprint). arXiv:1111.3349 Google Scholar
  13. 13.
    Pocchiola, M., Vegter, G.: Topologically sweeping visibility complexes via pseudotriangulations. Discret. Comput. Geom. 16(4), 419–453 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Reading, N.: Lattice congruences of the weak order. Order 21(4), 315–344 (2004/2005)Google Scholar
  15. 15.
    Reading, N.: Cambrian lattices. Adv. Math. 205(2), 313–353 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Reading, N.: Clusters, Coxeter-sortable elements and noncrossing partitions. Trans. Am. Math. Soc. 359(12), 5931–5958 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Reading, N.: Sortable elements and Cambrian lattices. Algebra Univers. 56(3–4), 411–437 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Rote, G., Santos, F., Streinu, I.: Pseudo-triangulations—a survey. In: Goodman, J.E., Pach, J., Pollack, R. (eds.) Surveys on Discrete and Computational Geometry. Contemporary Mathematics, vol. 453, pp. 343–410. American Mathematical Society, Providence (2008)CrossRefGoogle Scholar
  19. 19.
    Stein, W.A., et al.: Sage Mathematics Software (Version 4.8). The Sage Development Team. http://www.sagemath.org (2012)

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.CNRS & Laboratoire d’Informatique (LIX) École PolytechniquePalaiseau CedexFrance
  2. 2.Institut für Algebra, Zahlentheorie, Diskrete MathematikUniversität HannoverHannoverGermany

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