Combinatorial Hopf algebras and generalized Dehn-Sommerville relations

Compositio Mathematica

Volumn 142, Issue 1, 2006, Pages 1-30

  Aguiar, Marcelo ^a   Bergeron, Nantel ^b   Sottile, Frank ^a  

^a TEXAS A AND M UNIVERSITY   (United States)

^b YORK UNIVERSITY   (Canada)

Author keywords

Character; Eulerian poset; Generalized Dehn Sommerville relations; Hopf algebra; Non commutative symmetric function; Quasi symmetric function; Symmetric function

Indexed keywords

EID: 33747878960     PISSN: 0010437X     EISSN: None     Source Type: Journal    
DOI: 10.1112/S0010437X0500165X     Document Type: Review

References (35)

1. AFS00 M. Aguiar and W. Ferrer Santos, Galois connections for incidence Hopf algebras of partially ordered sets, Adv. Math. 151 (2000), 71-100.
2. Agu02 M. Aguiar, Infinitesimal Hopf algebra and the cd-index of poly topes, Discrete Comput. Geom. 27 (2002), 3-28.
3. Agu05 M. Aguiar, Universality of quasi-symmetric generating functions, in preparation (2005).
4. AS04 M. Aguiar and F. Sottile, Structure of the Loday-Ronco Hopf algebra of planar binary trees, J. Algebra, to appear.
5. AS05 M. Aguiar and F. Sottile, Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, Adv. Math. 191 (2005), 225-275.
6. BB85 M. Bayer and L. Billera, Generalized Dehn-Sommerville relations for polytopes, spheres, and Eulerian partially ordered sets, Invent. Math. 79 (1985), 143-157.
7. BMSW00 N. Bergeron, S. Mykytiuk, F. Sottile and S. van Willigenburg, Pieri operations on posets, J. Combin. Theory Ser. A 91 (2000), 84-110.
8. BMSW02 N. Bergeron, S. Mykytiuk, F. Sottile and S. van Willigenburg, Shifted quasi-symmetric functions and the Hopf algebra of peak functions, Discrete Math. 246 (2002), 57-66.
9. BS98 N. Bergeron and F. Sottile, Schubert polynomials, the Bruhat order, and the geometry of flag manifolds, Duke Math. J. 95 (1998), 373-423.
10. BS99 N. Bergeron and F. Sottile, Hopf algebras of edge-labeled posets, J. Algebra 216 (1999), 641-651.
11. BS02 N. Bergeron and F. Sottile, Skew Schubert functions and the Pieri formula for flag manifolds, Trans. Amer. Math. Soc. 354 (2002), 651-673.
12. BL00 L. Billera and N. Liu, Noncommutative enumeration in graded posets, J. Algebraic Combin. 12 (2000), 7-24.
13. BH95 S. Billey and M. Haiman, Schubert polynomials for the classical groups, J. Amer. Math. Soc. 8 (1995), 443-482.
14. Ehr96 R. Ehrenborg, On posets and Hopf algebras, Adv. Math. 119 (1996), 1-25.
15. Ges84 I. Gessel, Multipartite P-partitions and products of skew Schur functions, in Combinatorics and algebra, Boulder, CO, 1983, ed. C. Greene, Contemporary Mathematics, vol. 34 (American Mathematical Society, Providence, RI, 1984), 289-317.
16. GKLLRT95 I. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh and J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (1995), 218-348.
17. Haz03 M. Hazewinkel, Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions, Acta Appl. Math. 75 (2003), 55-83.
18. Hof00 M. E. Hoffman, Quasi-shuffle products, J. Algebraic Combin. 11 (2000), 49-68.
19. Hsi02 S. K. Hsiao, Structure of the peak Hopf algebra of quasi-symmetric functions, Preprint (2002).
20. JR79 S. Joni and G.-C. Rota, Coalgebras and bialgebras in combinatorics, Stud. Appl. Math. 61 (1979), 93-139.
21. LR98 J.-L. Loday and M. O. Ronco, Hopf algebra of the planar binary trees, Adv. Math. 139 (1998), 293-309.
22. LR02 J.-L. Loday and M. O. Ronco, Order structure on the algebra of permutations and of planar binary trees, J. Algebraic Combin. 15 (2002), 253-270.
23. Mac95 I. Macdonald, Symmetric functions and Hall polynomials, second edition (Oxford University Press, Oxford, 1995).
24. Mal94 C. Malvenuto, Produits et coproduits des fonctions quasi-symétriques et de l'algèbre des descents, no. 16 (Laboratoire de Combinatoire et d'Informatique Mathématique (LACIM), Université du Québec à Montréal, 1994).
25. MR95 C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (1995), 967-982.
26. MM65 J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211-264.
27. RotG4 G. C. Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 2 (1964), 340-368.
28. Sch87 W. R. Schmitt, Antipodes and incidence coalgebras, J. Combin. Theory Ser. A 46 (1987), 264-290.
29. Sch94 W. R. Schmitt, Incidence Hopf algebras, J. Pure Appl. Algebra 96 (1994), 299-330.
30. Slo94 N. J. A. Sloane, An on-line version of the encyclopedia of integer sequences, Electron. J. Combin. 1 (1994), http://akpublic.research.att.com/ ~njas/sequences/ol.html.
31. Sta72 R. Stanley, Ordered structures and partitions, Mem. Amer. Math. Soc. 119 (1972).
32. Sta95 R. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Adv. Math. 111 (1995), 166-194.
33. Sta97 R. Stanley, Enumerative combinatorics, vol. 1, with a foreword by G.-C. Rota, corrected reprint of the 1986 original (Cambridge University Press, Cambridge, 1997).
34. Sta99 R. Stanley, Enumerative combinatorics, vol. 2, with Appendix 1 by S. Fomin, Cambridge Studies in Advanced Mathematics, vol. 62 (Cambridge University Press, Cambridge, 1999).
35. Ste97 J. Stembridge, Enriched P-partitions, Trans. Amer. Math. Soc. 349 (1997), 763-788.