A decomposition of the group algebra of a hyperoctahedral group

TL;DR

This paper provides a detailed decomposition of the group algebra of a hyperoctahedral group, extending known results from symmetric groups and constructing explicit linear characters to realize the algebra's structure.

Contribution

It explicitly constructs the linear characters needed to decompose the hyperoctahedral group's algebra, answering Bonnafe's question affirmatively.

Findings

Decomposition of the hyperoctahedral group's algebra into induced linear characters.

Construction of explicit linear characters for the algebra.

Extension of Stanley's question to hyperoctahedral groups.

Abstract

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.