## Abstract

The peak set of a permutation σ is the set {*i* : σ(*i* − 1) < σ(*i*) > σ(*i* + 1)}. The group algebra of the symmetric group *S*_{n} admits a subalgebra in which elements are sums of permutations with a common descent set. In this paper we show the existence of a subalgebra of this descent algebra in which elements are sums of permutations sharing a common peak set. To prove the existence of this peak algebra we use the theory of enriched (*P*, γ)-partitions and the algebra of quasisymmetric peak functions studied by Stembridge (*Trans. Amer. Math. Soc*. 349 (1997) 763–788).

peaks Solomon's descent algebra quasisymmetric functions

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## References

- 1.D. Bayer and P. Diaconis, “Trailing the dovetail shuffle to its lair,”
*Ann. Appl. Probab*.**2**(2) (1992), 294–313.Google Scholar - 2.F. Bergeron and N. Bergeron, “Orthogonal idempotents in the descent algebra of
*B*_{n}and applications,”*J. Pure Appl. Algebra***79**(2) (1992), 109–129.Google Scholar - 3.F. Bergeron, N. Bergeron, R.B. Howlett, and D.E. Taylor, “A decomposition of the descent algebra of a finite Coxeter group,”
*J*.*Algebraic Combinatorics***1**(1992), 23–44.Google Scholar - 4.F. Bergeron, A. Garcia, and C. Reutenauer, “Homomorphisms between Solomon's descent algebra,”
*J. Algebra***150**(1992), 503–519.Google Scholar - 5.N. Bergeron, S. Mykytiuk, F. Sottile, and S. van Willigenburg, “Shifted quasisymmetric functions and the Hopf algebra of peak functions,”
*Discrete Math*.**246**(2002), 57–66.Google Scholar - 6.
- 7.P. Doyle and D. Rockmore, “Riffles, ruffles, and the turning algebra,” preprint.Google Scholar
- 8.A.M. Garsia and C. Reutenauer, “A decomposition of Solomon's descent algebra,”
*Adv. in Math*.**77**(1989), 189–262.Google Scholar - 9.I.M. Gessel, “Multipartite P-partitions and inner products of skew Schur functions,”
*Contemp. Math*.**34**(1984), 289–301.Google Scholar - 10.J.L. Loday,Opérations sur l'homologie cyclique des algèbre commutatives,
*Invent. Math*.**96**(1989), 205–230.Google Scholar - 11.C. Malvenuto and C. Reutenauer, “Duality between quasisymmetric functions and the Solomon descent algebra,”
*J. Algebra***177**(1995), 967–982.Google Scholar - 12.L. Solomon, “A Mackey formula in the group ring of a Coxeter group,”
*J. Algebra***41**(2) (1976), 255–268.Google Scholar - 13.R.P. Stanley,
*Enumerative Combinatorics*, Cambridge University Press, Cambridge, 1999, Vol.**2**.Google Scholar - 14.
- 15.J.R. Stembridge, “Enriched P-partitions,”
*Trans. Amer*.*Math. Soc*.**349**(1997), 763–788.Google Scholar

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