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Journal of Algebraic Combinatorics

, Volume 22, Issue 3, pp 343–375 | Cite as

Cyclic Descents and P-Partitions

  • T. Kyle PetersenEmail author
Article

Abstract

Louis Solomon showed that the group algebra of the symmetric group \(\mathfrak{S}_{n}\) n has a subalgebra called the descent algebra, generated by sums of permutations with a given descent set. In fact, he showed that every Coxeter group has something that can be called a descent algebra. There is also a commutative, semisimple subalgebra of Solomon's descent algebra generated by sums of permutations with the same number of descents: an “Eulerian” descent algebra. For any Coxeter group that is also a Weyl group, Paola Cellini proved the existence of a different Eulerian subalgebra based on a modified definition of descent. We derive the existence of Cellini's subalgebra for the case of the symmetric group and of the hyperoctahedral group using a variation on Richard Stanley's theory of P-partitions.

Keywords

descent algebra P-partition 

References

  1. 1.
    M. Aguiar, N. Bergeron, and K. Nyman, “The peak algebra and the descent algebras of types B and D,” Transactions of the Amer. Math. Soc. 356 (2004), 2781–2824.MathSciNetzbMATHGoogle Scholar
  2. 2.
    D. Bayer and P. Diaconis, “Trailing the dovetail shuffle to its lair,” Annals of Applied Probability 2 (1992), 294–313.MathSciNetzbMATHGoogle Scholar
  3. 3.
    N. Bergeron, “A decomposition of the descent algebra of the hyperoctahedral group. II.,” J. Algebra 148 (1992), 98–122.zbMATHMathSciNetGoogle Scholar
  4. 4.
    F. Bergeron and N. Bergeron, “A decomposition of the descent algebra of the hyperoctahedral group. I.,” J. Algebra 148 (1992), 86–97.MathSciNetzbMATHGoogle Scholar
  5. 5.
    F. Bergeron and N. Bergeron, “Orthogonal idempotents in the descent algebra of B n and applications,” J. Pure and Applied Algebra 79 (1992), 109–129.MathSciNetzbMATHGoogle Scholar
  6. 6.
    P. Cellini, “A general commutative descent algebra,” J. Algebra 175 (1995), 990–1014.zbMATHMathSciNetGoogle Scholar
  7. 7.
    P. Cellini, “A general commutative descent algebra. II. The case C n,” J. Algebra 175 (1995), 1015–1026.zbMATHMathSciNetGoogle Scholar
  8. 8.
    P. Cellini, “Cyclic Eulerian elements,” Europ. J. Combinatorics 19 (1998), 545–552.zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    C.-O. Chow, “Noncommutative symmetric functions of type B,” Ph.D. thesis (2001).Google Scholar
  10. 10.
    J. Fulman, “Affine shuffles, shuffles with cuts, the Whitehouse model, and patience sorting,” J. Algebra 231 (2000), 614–639.CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    J. Fulman, “Applications of the Brauer complex: card shuffling, permutation statistics, and dynamical systems,” J. Algebra 243 (2001), 96–122.CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    A. Garsia and C. Reutenauer, “A decomposition of Solomon's descent algebra,” Adv. Math. 77 (1989), 189–262.CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    I. Gessel, personal communication.Google Scholar
  14. 14.
    I. Gessel, “Multipartite P-partitions and inner products of skew Schur functions,” Contemporary Mathematics 34 (1984), 289–317.zbMATHMathSciNetGoogle Scholar
  15. 15.
    J.-L. Loday, “Operations sur l'homologie cyclique des algèbres commutatives,” Inventiones Mathematicae 96 (1989), 205–230.CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    K. Nyman, “The peak algebra of the symmetric group,” J. Algebraic Combin. 17 (2003), 309–322.CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    V. Reiner, “Quotients of Coxeter complexes and P-partitions,” Memoirs of the Amer. Math. Soc. 95 (1992).Google Scholar
  18. 18.
    V. Reiner, “Signed posets,” Journal of Combinatorial Theory Series A 62 (1993), 324–360.CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    L. Solomon, “A Mackey formula in the group ring of a finite Coxeter group,” J. Algebra 41 (1976), 255–264.zbMATHMathSciNetGoogle Scholar
  20. 20.
    R. Stanley, Enumerative Combinatorics, Volume I, Cambridge University Press, 1997.Google Scholar
  21. 21.
    J. Stembridge, “Enriched P-partitions,” Transactions of the Amer. Math. Soc. 349 (1997), 763–788.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of MathematicsBrandeis UniversityWalthamUSA

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