Geodesic
The peak algebra of the symmetric group
Journal of Algebraic Combinatorics, Tome 17 (2003) no. 3, pp. 309-322.

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Summary: The peak set of a permutation $sgr$ is the set $i : sgr( i - 1) sgr( i) > sgr( i + 1)$. 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 ( $P, gamma$)-partitions and the algebra of quasisymmetric peak functions studied by Stembridge ( Trans. Amer. Math. Soc. 349 (1997) 763-788).
Keywords: peaks, Solomon's descent algebra, quasisymmetric functions 
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Nyman, Kathryn L. The peak algebra of the symmetric group. Journal of Algebraic Combinatorics, Tome 17 (2003) no. 3, pp. 309-322. http://geodesic.mathdoc.fr/item/JAC_2003__17_3_a2/