  The descent algebra of a finite Coxeter group W is a subalgebra of the group
algebra defined by Solomon. Descent algebras of symmetric groups have
properties that are not shared by other Coxeter groups. For instance, the
natural map from the descent algebra of a symmetric group to its character ring
is a surjection with kernel equal the Jacobson radical. Thus, the descent
algebra implicitly encodes information about the representations of the
symmetric group, and a complete set of primitive idempotents in the character
ring leads to a decomposition of the group algebra into a sum of right ideals
indexed by partitions. Stanley asked whether this decomposition of the regular
representation of a symmetric group could be realized as a sum of
representations induced from linear characters of centralizers. This question
was answered positively by Bergeron, Bergeron, and Garsia, using a connection
with the free Lie algebra on n letters, and independently by Douglass,
Pfeiffer, and Roehrle, who connected the decomposition with the configuration
space of n-tuples of distinct complex numbers. The Mantaci-Reutenauer algebra
of a hyperoctahedral group is a subalgebra of the group algebra that contains
the descent algebra. Bonnafe and Hohlweg showed that the natural map from the
Mantaci-Reutenauer algebra to the character ring is a surjection with kernel
equal the Jacobson radical. In 2008, Bonnafe asked whether the analog to
Stanley's question about the decomposition of the group algebra into a sum of
induced linear characters holds. In this paper, we give a positive answer to
Bonnafe's question by explicitly constructing the required linear characters.
