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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
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References:
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