The Peak Algebra of the Symmetric Group
Kathryn L. Nyman
DOI: 10.1023/A:1025000905826
Abstract
The peak set of a permutation 
is the set { i : ( i - 1) <> ( i) > ( 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, 
)-partitions and the algebra of quasisymmetric peak functions studied by Stembridge ( Trans. Amer. Math. Soc. 349 (1997) 763-788).
is the set { i : ( i - 1) <> ( i) > ( 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, )-partitions and the algebra of quasisymmetric peak functions studied by Stembridge ( Trans. Amer. Math. Soc. 349 (1997) 763-788).Pages: 309–322
Keywords: peaks; Solomon's descent algebra; quasisymmetric functions
Full Text: PDF
References
1. D. Bayer and P. Diaconis, “Trailing the dovetail shuffle to its lair,” Ann. Appl. Probab. 2(2) (1992), 294-313.
2. F. Bergeron and N. Bergeron, “Orthogonal idempotents in the descent algebra of Bn and applications,” J. Pure Appl. Algebra 79(2) (1992), 109-129.
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.
4. F. Bergeron, A. Garcia, and C. Reutenauer, “Homomorphisms between Solomon's descent algebra,” J. Algebra 150 (1992), 503-519.
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.
6. P. Cellini, “A general commutative descent algebra,” J. Algebra 175 (1995), 990-1014.
7. P. Doyle and D. Rockmore, “Riffles, ruffles, and the turning algebra,” preprint.
8. A.M. Garsia and C. Reutenauer, “A decomposition of Solomon's descent algebra,” Adv. in Math. 77 (1989), 189-262. NYMAN
9. I.M. Gessel, “Multipartite P-partitions and inner products of skew Schur functions,” Contemp. Math. 34 (1984), 289-301.
10. J.L. Loday, Opérations sur l'homologie cyclique des alg`ebre commutatives, Invent. Math. 96 (1989), 205-230.
11. C. Malvenuto and C. Reutenauer, “Duality between quasisymmetric functions and the Solomon descent algebra,” J. Algebra 177 (1995), 967-982.
12. L. Solomon, “A Mackey formula in the group ring of a Coxeter group,” J. Algebra 41(2) (1976), 255-268.
13. R.P. Stanley, Enumerative Combinatorics, Cambridge University Press, Cambridge, 1999, Vol. 2.
14. R.P. Stanley, “Ordered structures and partitions,” Mem. Amer. Math. Soc. 119 (1972).
15. J.R. Stembridge, “Enriched P-partitions,” Trans. Amer. Math. Soc. 349 (1997), 763-788.
2. F. Bergeron and N. Bergeron, “Orthogonal idempotents in the descent algebra of Bn and applications,” J. Pure Appl. Algebra 79(2) (1992), 109-129.
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.
4. F. Bergeron, A. Garcia, and C. Reutenauer, “Homomorphisms between Solomon's descent algebra,” J. Algebra 150 (1992), 503-519.
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.
6. P. Cellini, “A general commutative descent algebra,” J. Algebra 175 (1995), 990-1014.
7. P. Doyle and D. Rockmore, “Riffles, ruffles, and the turning algebra,” preprint.
8. A.M. Garsia and C. Reutenauer, “A decomposition of Solomon's descent algebra,” Adv. in Math. 77 (1989), 189-262. NYMAN
9. I.M. Gessel, “Multipartite P-partitions and inner products of skew Schur functions,” Contemp. Math. 34 (1984), 289-301.
10. J.L. Loday, Opérations sur l'homologie cyclique des alg`ebre commutatives, Invent. Math. 96 (1989), 205-230.
11. C. Malvenuto and C. Reutenauer, “Duality between quasisymmetric functions and the Solomon descent algebra,” J. Algebra 177 (1995), 967-982.
12. L. Solomon, “A Mackey formula in the group ring of a Coxeter group,” J. Algebra 41(2) (1976), 255-268.
13. R.P. Stanley, Enumerative Combinatorics, Cambridge University Press, Cambridge, 1999, Vol. 2.
14. R.P. Stanley, “Ordered structures and partitions,” Mem. Amer. Math. Soc. 119 (1972).
15. J.R. Stembridge, “Enriched P-partitions,” Trans. Amer. Math. Soc. 349 (1997), 763-788.