Balanced and Bruhat Graphs

Abstract

We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an R-labeling, imply the existence of the (non-homogeneous) \(\mathbf{c}\mathbf{d}\)-index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian posets, we show an analogue of Alexander duality for bounded balanced digraphs. For Bruhat graphs of Coxeter groups, an important family of balanced graphs, our theory gives elementary proofs of the existence of the complete \(\mathbf{c}\mathbf{d}\)-index and its properties. We also introduce the rising and falling quasisymmetric functions of a labeled acyclic digraph and show they are Hopf algebra homomorphisms mapping balanced digraphs to the Stembridge peak algebra. We conjecture non-negativity of the \(\mathbf{c}\mathbf{d}\)-index for acyclic digraphs having a balanced linear edge labeling.

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References

  1. 1.

    M. Bayer and L. Billera, Generalized Dehn–Sommerville relations for polytopes, spheres and Eulerian partially ordered sets, Invent. Math. 79 (1985), 143–157

    MathSciNet  Article  Google Scholar 

  2. 2.

    M. Bayer and R. Ehrenborg, The toric \(h\)-vectors of partially ordered sets, Trans. Amer. Math. Soc. 352 (2000), 4515–4531

    MathSciNet  Article  Google Scholar 

  3. 3.

    M. M. Bayer and G. Hetyei, Flag vectors of Eulerian partially ordered sets, European J. Combin. 22 (2001), 5–26

    MathSciNet  Article  Google Scholar 

  4. 4.

    M. Bayer and A. Klapper, A new index for polytopes, Discrete Comput. Geom. 6 (1991), 33–47

    MathSciNet  Article  Google Scholar 

  5. 5.

    N. Bergeron, S. Mykytiuk, F. Sottile and S. van Willigenburg, Noncommutative Pieri operators on posets, J. Combin. Theory Ser. A 91 (2000), 84–110

    MathSciNet  Article  Google Scholar 

  6. 6.

    N. Bergeron and F. Sottile, Hopf algebra and edge-labeled posets, J. of Alg. 216 (1999), 641–651

    MathSciNet  Article  Google Scholar 

  7. 7.

    L. J. Billera and F. Brenti, Quasisymmetric functions and Kazhdan–Lusztig polynomials, Israel Jour. Math. 184 (2011), 317–348

    MathSciNet  Article  Google Scholar 

  8. 8.

    L. J. Billera and R. Ehrenborg, Monotonicity of the cd-index for polytopes, Math. Z. 233 (2000), 421–441

    MathSciNet  Article  Google Scholar 

  9. 9.

    L. J. Billera, R. Ehrenborg, and M. Readdy, The c-2d-index of oriented matroids, J. Combin. Theory Ser. A 80 (1997), 79–105

  10. 10.

    L. J. Billera, R. Ehrenborg, and M. Readdy, The \({{\bf c}}{{\bf d}}\)-index of zonotopes and arrangements, Mathematical essays in honor of Gian-Carlo Rota (B. Sagan and R. P. Stanley, eds.), Birkhäuser, Boston, 1998, 23–40.

  11. 11.

    L. J. Billera and N. Liu, Noncommutative enumeration in graded posets, J. Algebraic Combin. 12 (2000), 7–24

    MathSciNet  Article  Google Scholar 

  12. 12.

    A. Björner, Shellable and Cohen–Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), 159–183

  13. 13.

    A. Björner and F. Brenti, “Combinatorics of Coxeter groups,” Springer, 2005.

  14. 14.

    A. Björner and M. Wachs, Shellable nonpure complexes and posets. I., Trans. Amer. Math. Soc. 348 (1996), 1299–1327

  15. 15.

    S. A. Blanco, The complete \({{\bf c}}{{\bf d}}\)-index of dihedral and universal Coxeter groups, Electron. J. Combin. 18 (2011), no 1, Paper 174, 16pp.

  16. 16.

    S. A. Blanco, Shortest path poset of Bruhat intervals, J. Algebraic Combin. 38 (2013), 585–596

    MathSciNet  Article  Google Scholar 

  17. 17.

    F. Brenti, Lattice paths and Kazhdan–Lusztig polynomials, Jour. Amer. Math. Soc. 11 (1998), 229–259

    MathSciNet  Article  Google Scholar 

  18. 18.

    F. Brenti and F. Caselli, Peak algebras, paths in the Bruhat graph and Kazhdan–Lusztig polynmials, Adv. in Math 304 (2017), 539–582

    MathSciNet  Article  Google Scholar 

  19. 19.

    F. Brenti, F. Caselli and M. Marietti, Special Matchings and Coxeter groups, Adv. Applied Math 38 (2007), 210–226

    Article  Google Scholar 

  20. 20.

    M. J. Dyer, “Hecke algebras and reflections in Coxeter groups,” Doctoral dissertation, University of Sydney, 1987.

  21. 21.

    M. J. Dyer, Hecke algebras and shellings of Bruhat intervals, Compositio Math. 89 (1993), 91–115

    MathSciNet  MATH  Google Scholar 

  22. 22.

    R. Ehrenborg, On posets and Hopf algebras, Adv. Math. 119 (1996), 1–25

    MathSciNet  Article  Google Scholar 

  23. 23.

    R. Ehrenborg, Lifting inequalities for polytopes, Adv. Math. 193 (2005), 205–222

    MathSciNet  Article  Google Scholar 

  24. 24.

    R. Ehrenborg, Inequalities for zonotopes, in MSRI Publication on Combinatorial and Computational Geometry (J.E. Goodman, J. Pach and E. Eelzl, eds.), Cambridge University Press, Cambridge, England, 2005, pp. 277–286.

  25. 25.

    R. Ehrenborg, M. Goresky and M. Readdy, Euler flag enumeration of Whitney stratified spaces, Adv. Math. 268 (2015), 85–128

    MathSciNet  Article  Google Scholar 

  26. 26.

    R. Ehrenborg, G. Hetyei and M. Readdy, Level Eulerian posets, Graphs and Combinatorics 29 (2013), 857–882

    MathSciNet  Article  Google Scholar 

  27. 27.

    R. Ehrenborg and K. Karu, Decomposition theorem for the \({{\bf c}}{{\bf d}}\)-index of Gorenstein* posets, J. Algebraic Combin. 26 (2007), 225–251

  28. 28.

    R. Ehrenborg and M. Readdy, Sheffer posets and \(r\)-signed permutations, Ann. Sci. Math. Québec 19 (1995), 173–196

    MathSciNet  MATH  Google Scholar 

  29. 29.

    R. Ehrenborg and M. Readdy, The r-cubical lattice and a generalization of the cd-index, European J. Combin. 17 (1996), 709–725

  30. 30.

    R. Ehrenborg and M. Readdy, Coproducts and the \({{\bf c}}{{\bf d}}\)-index, J. Algebraic Combin. 8 (1998), 273–299

  31. 31.

    R. Ehrenborg and M. Readdy, Homology of Newtonian coalgebras, European J. Combin. 23 (2002), 919–927

    MathSciNet  Article  Google Scholar 

  32. 32.

    R. Ehrenborg and M. Readdy, On the non-existence of an \(R\)-labeling, Order 28 (2011), 437–442

    MathSciNet  Article  Google Scholar 

  33. 33.

    R. Ehrenborg and M. Readdy, Manifold arrangements, J. Combin. Theory Ser. A 125 (2014), 214–239

    MathSciNet  Article  Google Scholar 

  34. 34.

    R. Ehrenborg and M. Readdy, The Tchebyshev transforms of the first and second kind, Ann. Comb. 14 (2010), 211–244

    MathSciNet  Article  Google Scholar 

  35. 35.

    R. Ehrenborg, M. Readdy and M. Slone, Affine and toric hyperplane arrangements, Discrete Comput. Geom. 41 (2009), 481–512

    MathSciNet  Article  Google Scholar 

  36. 36.

    N. J. Y. Fan and L. He, The complete \({{\bf c}}{{\bf d}}\)-index of Boolean lattices, Electron. J. Combin. 22 (2015), no. 2, Paper 2.45, 18 pp.

  37. 37.

    N. J. Y. Fan and L. He, On the non-negativity of the complete \({{\bf c}}{{\bf d}}\)-index, Discrete Math. 338 (2015), 2037–2041

  38. 38.

    N. B. Fox, A lattice path interpretation of the diamond product, Ann. Comb. 20 (2016), 569–586

    MathSciNet  Article  Google Scholar 

  39. 39.

    S. A. Joni and G.-C. Rota, Coalgebras and bialgebras in combinatorics, Stud. Appl. Math. 61 (1979), 93–139

    MathSciNet  Article  Google Scholar 

  40. 40.

    K. Karu, Hard Lefschetz theorem for nonrational polytopes, Invent. Math. 157 (2004), 419–447

    MathSciNet  Article  Google Scholar 

  41. 41.

    K. Karu, The \(cd\)-index of fans and posets, Compos. Math. 142 (2006), 701–718

    MathSciNet  Article  Google Scholar 

  42. 42.

    K. Karu, On the complete \({\bf cd}\)-index of a Bruhat interval, J. Algebraic Combin. 38 (2013), 27–541

    MathSciNet  Article  Google Scholar 

  43. 43.

    D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math 53 (1979), 165–184

    MathSciNet  Article  Google Scholar 

  44. 44.

    D. Kazhdan and G. Lusztig, Schubert varieties and Poincaré duality, Proc. Sympos. Pure Math. 34 (1980), 185–203

    Article  Google Scholar 

  45. 45.

    C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (1995), 967–982

    MathSciNet  Article  Google Scholar 

  46. 46.

    S. Morel, Note sur les polynômes de Kazhdan–Lusztig, Math. Z 268 (2011), 593–600

    MathSciNet  Article  Google Scholar 

  47. 47.

    N. Reading, The \(cd\)-index of Bruhat intervals, Electron. J. Combin. 11 (2004), no. 1, Research Paper 74, 25 pp.

  48. 48.

    M. Slone, “Homological combinatorics and extensions of the \({\bf cd}\)-index,” Doctoral dissertation, University of Kentucky, 2008.

  49. 49.

    R. P. Stanley, Flag \(f\)-vectors and the \(cd\)-index, Math. Z. 216 (1994), 483–499

    MathSciNet  Article  Google Scholar 

  50. 50.

    R. P. Stanley, “Enumerative combinatorics. Volume 1. Second edition.,” Cambridge University Press, Cambridge, 2012.

  51. 51.

    R. P. Stanley, Flag \(f\)-vectors and the \(cd\)-index, Math. Z. 216 (1994), 483–499

  52. 52.

    J. Stembridge, Enriched \(P\)-partitions, Trans. Amer. Math. Soc. 349 (1997), 763–788

    MathSciNet  Article  Google Scholar 

  53. 53.

    M. Sweedler, “Hopf Algebras,” Benjamin, New York, 1969.

  54. 54.

    D.-N. Verma, Möbius inversion for the Bruhat order on a Weyl group, Ann. Sci. École Norm. Sup. 4 (1971), 393–398

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors thank the referee for helpful comments. The first author was partially supported by National Science Foundation Grant 0902063. This work was partially supported by grants from the Simons Foundation (#429370 to Richard Ehrenborg; #206001 and #422467 to Margaret Readdy). Both authors would like to thank the Princeton University Mathematics Department for its hospitality and support during the academic year 2014–2015, and the Institute for Advanced Study for hosting a research visit in Summer 2019.

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Correspondence to Richard Ehrenborg.

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Ehrenborg, R., Readdy, M. Balanced and Bruhat Graphs. Ann. Comb. 24, 587–617 (2020). https://doi.org/10.1007/s00026-020-00510-7

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Keywords

  • Alexander duality
  • Balanced digraph
  • Bruhat graph
  • \(\mathbf{c}\mathbf{d}\)-Index
  • Eulerian poset
  • Quasisymmetric function
  • R-labeling

Mathematics Subject Classification

  • Primary 06A11
  • 52B05
  • Secondary 05E05
  • 06A08
  • 16T15
  • 20F55