Order Structure on the Algebra of Permutations and of Planar Binary Trees
Jean-Louis Loday
and María O. Ronco
DOI: 10.1023/A:1015064508594
Abstract
Let X n be either the symmetric group on n letters, the set of planar binary n-trees or the set of vertices of the ( n - 1)-dimensional cube. In each case there exists a graded associative product on 
n 
0 K[ X n]. We prove that it can be described explicitly by using the weak Bruhat order on S n, the left-to-right order on planar trees, the lexicographic order in the cube case.
n 0 K[ X n]. We prove that it can be described explicitly by using the weak Bruhat order on S n, the left-to-right order on planar trees, the lexicographic order in the cube case.Pages: 253–270
Keywords: planar binary tree; order structure; weak Bruhat order; algebra of permutations; dendriform algebra
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References
1. N. Bourbaki, Groupes et alg`ebres de Lie, Hermann Paris 1968; reprinted Masson, Paris, 1981, Ch. 4-6.
2. Chr. Brouder and A. Frabetti, “Renormalization of QED with trees,” Eur. Phys. J. C 19 (2001), 715-741.
3. I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh, and J.-Y. Thibon, “Noncommutative symmetric functions,” Adv. Math. 112(2) (1995), 218-348.
4. J.-L. Loday, “Dialgebras,” in Dialgebras and Related Operads, Springer Lecture Notes in Math., Vol. 1763, (2001), 7-66.
5. J.-L. Loday and M.O. Ronco, “Hopf algebra of the planar binary trees,” Adv. Math. 139(2) (1998), 293-309.
6. C. Malvenuto and Chr. Reutenauer, “Duality between quasi-symmetric functions and the Solomon descent algebra,” J. Algebra 177 (1995), 967-982.
7. L. Solomon, “A Mackey formula in the group ring of a Coxeter group,” J. Algebra 41 (1976), 255-264.
8. R.P. Stanley, Enumerative Combinatorics, Vol. I, The Wadsworth and Brooks/Cole Mathematics Series, 1986.
2. Chr. Brouder and A. Frabetti, “Renormalization of QED with trees,” Eur. Phys. J. C 19 (2001), 715-741.
3. I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh, and J.-Y. Thibon, “Noncommutative symmetric functions,” Adv. Math. 112(2) (1995), 218-348.
4. J.-L. Loday, “Dialgebras,” in Dialgebras and Related Operads, Springer Lecture Notes in Math., Vol. 1763, (2001), 7-66.
5. J.-L. Loday and M.O. Ronco, “Hopf algebra of the planar binary trees,” Adv. Math. 139(2) (1998), 293-309.
6. C. Malvenuto and Chr. Reutenauer, “Duality between quasi-symmetric functions and the Solomon descent algebra,” J. Algebra 177 (1995), 967-982.
7. L. Solomon, “A Mackey formula in the group ring of a Coxeter group,” J. Algebra 41 (1976), 255-264.
8. R.P. Stanley, Enumerative Combinatorics, Vol. I, The Wadsworth and Brooks/Cole Mathematics Series, 1986.