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A decomposition of the descent algebra of a finite Coxeter group.(English)Zbl 0798.20031

This paper unifies previous work of A. Garcia and the reviewer, and of the two first authors, which gave the primitive idempotents of the descent algebra of the symmetric group and of the hyperoctahedral group [Adv. Math. 77, No. 2, 189-262 (1989;Zbl 0716.20006), J. Algebra 148, 86-97 and 98-122 (1992;Zbl 0798.20008)]. These idempotents are constructed in a uniform way for each finite Coxeter group \(W\). They are lifted from the parabolic Burnside algebra. The dimensions of the corresponding representations of \(W\) are given, the radical of the descent algebra is determined and the link with the exponents of \(W\) is given.

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
19A22 Frobenius induction, Burnside and representation rings
20C30 Representations of finite symmetric groups

Cite

References:

[1]J.L. Alperin, Local Representation Theory, New York, Cambridge University Press, 1986. ·Zbl 0593.20003
[2]F. Bergeron and N. Bergeron, “A decomposition of the descent algebra of the hyperoctahedral group I,” Journal of Algebra (accepted 1990). ·Zbl 0798.20008
[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). ·Zbl 0790.20058
[4]F. Bergeron, A. Garsia, and C. Reutenauer, “Homomorphisms between Solomon”s descent algebras,” submitted. ·Zbl 0770.20008
[5]N. Bergeron, “A decomposition of the descent algebra of the hyperoctahedral group II,” Journal of Algebra (accepted 1990). ·Zbl 0798.20008
[6]N. Bourbaki, Groupes et Algèbres de Lie, Chapitres 4, 5 et 6, Éléments de Mathématiques, New York, Masson, 1981. ·Zbl 0483.22001
[7]R.W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, New York, Wiley, 1985. ·Zbl 0567.20023
[8]R.W. Carter, “Conjugacy classes in the Weyl group,” Compositio Mathematics, vol. 25, 1972, pp. 1-59. ·Zbl 0254.17005
[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. ·Zbl 0709.17003
[10]A. Garsia and C. Reutenauer, “A decomposition of Solomon”s descent algebras,“ <Emphasis Type=”Italic”>Advances in Mathematics, vol. 77, no. 2, pp. 189-262, 1989. ·Zbl 0716.20006
[11]R. B. Howlett, “Normalizers of parabolic subgroups of reflection groups,” Journal of the London Mathematics Society, vol. 21, pp. 62-80, 1980. ·Zbl 0427.20040
[12]G. C. Shepahard and J. A. Todd, “Finite unitary reflection groups,” Canadian Journal of Mathematics, vol. 6, pp. 274-304, 1954. ·Zbl 0055.14305
[13]L. Solomon, “Invariants of finite reflection groups,” Nagoya Mathematics, vol. 22, pp. 57-64, 1963. ·Zbl 0117.27104
[14]L. Solomon, “A Mackey formula in the group ring of a Coxeter group,” Journal of Algebra, vol. 41, pp. 255-264, 1976. ·Zbl 0355.20007
[15]R. Steinberg, “Endomorphisms of Linear Algebraic groups,” Mem. American Mathematics Society, vol. 80, 1968, pp. 1-108. ·Zbl 0164.02902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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