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F. Bergeron , N. Bergeron , R.B. Howlett and D.E. Taylor

The purpose of this paper is twofold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see [2], [3], [4], [5] and [10]) on the structure of the descent algebras of the Coxeter groups of type A _n and B _n. But we shall also extend these results to the descent algebra of an arbitrary finite Coxeter group W. The descent algebra, introduced by Solomon in [14], is a subalgebra of the group algebra of W. It is closely related to the subring of the Burnside ring B( W) spanned by the permutation representations W/W _J, where the W _J are the parabolic subgroups of W. Specifically, our purpose is to lift a basis of primitive idempotents of the parabolic Burnside algebra to a basis of idempotents of the descent algebra.

1. J.L. Alperin, Local Representation Theory, New York, Cambridge University Press, 1986.
2. F. Bergeron and N. Bergeron, "A decomposition of the descent algebra of the hyperoctahedral group I," Journal of Algebra (accepted 1990).
3. F. Bergeron and N. Bergeron, "Symbolic manipulation for the study of the descent algebra of finite Coxeter groups," Journal of Symbolic Computation (accepted 1990).
4. F. Bergeron, A. Garsia, and C. Reutenauer, "Homomorphisms between Solomon's descent algebras," submitted.
5. N. Bergeron, "A decomposition of the descent algebra of the hyperoctahedral group II," Journal of Algebra (accepted 1990).
6. N. Bourbaki, Groupes et Algebres de Lie, Chapitres 4, 5 et 6, Elements de Mathematiques, New York, Masson, 1981.
7. R.W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, New York, Wiley, 1985.
8. R.W. Carter, "Conjugacy classes in the Weyl group," Compositio Mathematics, vol. 25, 1972, pp. 1-59.
9. A.M. Garsia, Combinatorics of the free Lie algebra and the symmetric group, Analysis, et cetera, Research papers published in honor of Jurgen Moser's 60th birthday, ed: Paul H. Rabinowitz and Eduard Zehnder, Academic Press, 1990.
10. A. Garsia and C. Reutenauer, "A decomposition of Solomon's descent algebras," Advances in Mathematics, vol. 77, no. 2, pp. 189-262, 1989.
11. R. B. Howlett, "Normalizers of parabolic subgroups of reflection groups," Journal of the London Mathematics Society, vol. 21, pp. 62-80, 1980.
12. G. C. Shepahard and J. A. Todd, "Finite unitary reflection groups," Canadian Journal of Mathematics, vol. 6, pp. 274-304, 1954.
13. L. Solomon, "Invariants of finite reflection groups," Nagcya Mathematics, vol. 22, pp. 57-64, 1963.
14. L. Solomon, "A Mackey formula in the group ring of a Coxeter group," Journal of Algebra, vol. 41, pp. 255-264, 1976.
15. R. Steinberg, "Endomorphisms of Linear Algebraic groups," Mem. American Mathematics Society, vol. 80, 1968, pp. 1-108.