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Journal of Algebraic Combinatorics

, Volume 32, Issue 4, pp 597–627 | Cite as

Geometric combinatorial algebras: cyclohedron and simplex

  • S. ForceyEmail author
  • Derriell Springfield
Article

Abstract

In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the Malvenuto–Reutenauer algebra of permutations and the Loday–Ronco algebra of binary trees. Connecting algebra maps arise from a new generalization of the Tonks projection from the permutohedron to the associahedron, which we discover via the viewpoint of the graph associahedra of Carr and Devadoss. At the same time, that viewpoint allows exciting geometrical insights into the multiplicative structure of the algebras involved. Extending the Tonks projection also reveals a new graded algebra structure on the simplices. Finally this latter is extended to a new graded Hopf algebra with basis all the faces of the simplices.

Keywords

Hopf algebra Graph associahedron Cyclohedron Graded algebra 

References

  1. 1.
    Aguiar, M., Sottile, F.: Structure of the Malvenuto–Reutenauer Hopf algebra of permutations. Adv. Math. 191(2), 225–275 (2005). MR2103213 (2005m:05226) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aguiar, M., Sottile, F.: Structure of the Loday–Ronco Hopf algebra of trees. J. Algebra 295(2), 473–511 (2006). MR2194965 (2006k:16078) zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bott, R., Taubes, C.: On the self-linking of knots. J. Math. Phys. 35(10), 5247–5287 (1994). Topology and physics. MR1295465 (95g:57008) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Carr, M.P., Devadoss, S.L.: Coxeter complexes and graph-associahedra. Topol. Appl. 153(1–2), 2155–2168 (2006). MR2239078 (2007c:52012) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chapoton, F.: Bigèbres différentielles graduées associées aux permutoèdres, associaèdres et hypercubes. Ann. Inst. Fourier (Grenoble) 50(4), 1127–1153 (2000). MR1799740 (2002f:16081) zbMATHMathSciNetGoogle Scholar
  6. 6.
    Devadoss, S., Forcey, S.: Marked tubes and the graph multiplihedron. Algebraic Geom. Topol. 8(4), 2081–2108 (2008). MR2460880 zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Devadoss, S.L.: A space of cyclohedra. Discrete Comput. Geom. 29(1), 61–75 (2003). MR1946794 (2003j:57027) zbMATHMathSciNetGoogle Scholar
  8. 8.
    Devadoss, S.L.: A realization of graph associahedra. Discrete Math. 309(1), 271–276 (2009). MR2479448 zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Forcey, S., Lauve, A., Sottile, F.: Constructing cofree compositional coalgebras (manuscript in preparation) Google Scholar
  10. 10.
    Hohlweg, C., Lange, C.E.M.C.: Realizations of the associahedron and cyclohedron. Discrete Comput. Geom. 37(4), 517–543 (2007). MR2321739 (2008g:52021) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Loday, J.-L., Ronco, M.O.: Hopf algebra of the planar binary trees. Adv. Math. 139(2), 293–309 (1998). MR1654173 (99m:16063) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Loday, J.-L., Ronco, M.O.: Trialgebras and families of polytopes. In: Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology and Algebraic K-Theory. Contemp. Math., vol. 346, pp. 369–398. Am. Math. Soc., Providence (2004). MR2066507 (2006e:18016) Google Scholar
  13. 13.
    Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177(3), 967–982 (1995). MR1358493 (97d:05277) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Markl, M.: Simplex, associahedron, and cyclohedron. In: Higher Homotopy Structures in Topology and Mathematical Physics, Poughkeepsie, NY, 1996. Contemp. Math., vol. 227, pp. 235–265. Am. Math. Soc., Providence (1999). MR1665469 (99m:57020) Google Scholar
  15. 15.
    Morton, J., Shiu, A., Pachter, L., Sturmfels, B.: The cyclohedron test for finding periodic genes in time course expression studies. Stat. Appl. Genet. Mol. Biol. 6 (2007), Art. 21, 25 pp. (electronic). MR2349914 Google Scholar
  16. 16.
    Postnikov, A., Reiner, V., Williams, L.: Faces of generalized permutohedra. Doc. Math. 13, 207–273 (2008). MR2520477 zbMATHMathSciNetGoogle Scholar
  17. 17.
    Postnikov, A.: Permutohedra, associahedra, and beyond. Int. Math. Res. Not. IMRN 6, 1026–1106 (2009). MR2487491 MathSciNetGoogle Scholar
  18. 18.
    Reading, N.: Cambrian lattices. Adv. Math. 205(2), 313–353 (2006). MR2258260 (2007g:05195) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Simion, R.: A type-B associahedron. Adv. Appl. Math. 30(1–2), 2–25 (2003). Formal power series and algebraic combinatorics (Scottsdale, AZ, 2001). MR1979780 (2004h:52013) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Tonks, A.: Relating the associahedron and the permutohedron. In: Operads: Proceedings of Renaissance Conferences, Hartford, CT/Luminy, 1995. Contemp. Math., vol. 202, pp. 33–36. Am. Math. Soc., Providence (1997). MR1436915 (98c:52015) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Tennessee State UniversityNashvilleUSA
  2. 2.The University of AkronAkronUSA

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