Orthogonal idempotents in the descent algebra of \(B_ n\) and applications
Let \(B_ n\) denote the hyperoctahedral group and \(\Sigma(B_ n)\) be its descent algebra. The authors consider some idempotents \(\rho^ k_ n\), \(1 \leq k \leq n\), in \(\Sigma(B_ n)\) and show that the dimensions of the corresponding right ideals \(\mathbb{Q}[B_ n]\rho^ k_ n\) are the numbers of elements of \(B_ n\) having \(k\) positive cycles, respectively. They also prove that the algebra \(\Lambda\) generated by the classes of elements of \(B_ n\) having equal numbers of descents is commutative. They show that they can define some \(\lambda\) operators in \(\Lambda\) which commute with the Hochschild boundary operators, providing a decomposition for the commutative hyperoctahedral algebra homology.
commutative hyperoctahedral algebra homology
scientific article; zbMATH DE number 89565