Lie Representations and an Algebra Containing Solomon's
Frédéric Patras
and Christophe Reutenauer
DOI: 10.1023/A:1021856522624
Abstract
We introduce and study a Hopf algebra containing the descent algebra as a sub-Hopf-algebra. It has the main algebraic properties of the descent algebra, and more: it is a sub-Hopf-algebra of the direct sum of the symmetric group algebras; it is closed under the corresponding inner product; it is cocommutative, so it is an enveloping algebra; it contains all Lie idempotents of the symmetric group algebras. Moreover, its primitive elements are exactly the Lie elements which lie in the symmetric group algebras.
Pages: 301–314
Keywords: descent algebra; Hopf algebra; Lie idempotent; symmetric group algebras; quasi-symmetric functions; Lie elements
Full Text: PDF
References
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2. G. Duchamp, “Orthogonal projection onto the free Lie algebra,” Theor. Comput. Sci. 79(1) (1991), 227-239.
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. I.M. Gessel, “Multipartite P-partitions and inner products of skew Schur functions,” in Combinatorics and algebra, Boulder, CO., 1983, Contemp. Math., Vol. 34, Amer. Math. Soc., Providence, R.I., 1984, pp. 289-317.
5. A. J\ddot ollenbeck, “Nichtkommutative Charaktertheorie der symmetrischen Gruppen,” Bayreuth. Math. Schr. 56 (1999), 1-41.
6. D. Krob, B. Leclerc, and J.-Y. Thibon, “Noncommutative symmetric functions. II: Transformations of alphabets,” Int. J. Algebra Comput. 7(2) (1997), 181-264.
7. C. Malvenuto and C. Reutenauer, “Duality between quasi-symmetric functions and the Solomon descent algebra,” J. Algebra 177(3) (1995), 967-982.
8. J.W. Milnor and J.C. Moore, “On the structure of Hopf algebras,” Ann. Math. Ser. II. 81 (1965), 211-264.
9. F. Patras, “L'alg`ebre des descentes d'une big`ebre graduée,” J. Algebra. 170(2) (1994), 547-566.
10. F. Patras and C. Reutenauer, “Higher Lie idempotents,” J. Algebra 222(1) (1999), 51-64.
11. S. Poirier and C. Reutenauer, “Alg`ebres de Hopf de tableaux,” Ann. Sci. Math. Québec 19(1) (1995), 79-90.
12. C. Reutenauer, “Dimensions and characters of the derived series of the free Lie algebra,” in Mots, Mélanges offerts `a Sch\ddot utzenberger, Herm`es, 1990, pp. 171-184.
13. C. Reutenauer, Free Lie algebras, Oxford University Press, 1993.
14. M. Schocker, “ \ddot Uber die h\ddot oheren Lie-Darstellungen der symmetrischen Gruppen,” Ph.D. Thesis, Kiel, 2001.
15. L. Solomon, “A Mackey formula in the group algebra of a finite Coxeter group,” J. Algebra 41 (1976), 255-268.
16. M. Sweedler, Hopf Algebras, Benjamin, New York, 1969.