Geometric combinatorial algebras: cyclohedron and simplex
S. Forcey
and Derriell Springfield
The University of Akron, 302 Buchtel Common, Akron, OH 44325, USA
DOI: 10.1007/s10801-010-0229-5
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.
Pages: 597–627
Keywords: keywords Hopf algebra; graph associahedron; cyclohedron; graded algebra
Full Text: PDF
References
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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)
4. Carr, M.P., Devadoss, S.L.: Coxeter complexes and graph-associahedra. Topol. Appl. 153(1-2), 2155- 2168 (2006). MR2239078 (2007c:52012)
5. Chapoton, F.: Bigèbres différentielles graduées associées aux permutoèdres, associaèdres et hypercubes. Ann. Inst. Fourier (Grenoble) 50(4), 1127-1153 (2000). MR1799740 (2002f:16081)
6. Devadoss, S., Forcey, S.: Marked tubes and the graph multiplihedron. Algebraic Geom. Topol. 8(4), 2081-2108 (2008). MR2460880
7. Devadoss, S.L.: A space of cyclohedra. Discrete Comput. Geom. 29(1), 61-75 (2003). MR1946794 (2003j:57027)
8. Devadoss, S.L.: A realization of graph associahedra. Discrete Math. 309(1), 271-276 (2009). MR2479448
9. Forcey, S., Lauve, A., Sottile, F.: Constructing cofree compositional coalgebras (manuscript in preparation)
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)
11. Loday, J.-L., Ronco, M.O.: Hopf algebra of the planar binary trees. Adv. Math. 139(2), 293-309 (1998). MR1654173 (99m:16063)
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)
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)
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)
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
16. Postnikov, A., Reiner, V., Williams, L.: Faces of generalized permutohedra. Doc. Math. 13, 207-273 (2008). MR2520477
17. Postnikov, A.: Permutohedra, associahedra, and beyond. Int. Math. Res. Not. IMRN 6, 1026-1106 (2009). MR2487491
18. Reading, N.: Cambrian lattices. Adv. Math. 205(2), 313-353 (2006). MR2258260 (2007g:05195)
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)
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.
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