Hyperoctahedral operations on Hochschild homology
The author constructs a doubly-parametrized family of idempotents \(\{\rho^{(\ell,k)}_n\}\) in the group algebra over the hyperoctahedral group, \(\mathbb{Q}[B_n]\), which appears to be the right hyperoctahedral analogue of the family of ``eulerian'' idempotents \(\{e^i_n\}\) in \(\mathbb{Q}[S_n]\) [see \textit{C. Reutenauer}, Free Lie algebras. (Oxf. Univ. Press, 1993; Zbl 0798.17001)]. Namely, those idempotents commute with the Hochschild abstract boundary and are projections onto some hyperoctahedral symmetric powers of the free Lie algebra (theorem 4.1); the characters of the corresponding representations are computed by means of induced characters in section 7. Moreover, since \((e^n_n,B_n)\) is a Gelfand pair, the idempotents \(\{\rho^{(\ell,k)}_n\}_{\ell+k=n}\) are the minimal idempotents of \(e^n_n\mathbb{Q} [B_n]\) (corollary 5.3), from which it follows that the idempotents \(\{\rho^{(\ell,k)}_n\}\) do give the finest decomposition of the Hochschild homology with hyperoctahedral structure (theorem 5.5). The corresponding components of the Hochschild homology may be computed by means of hyperoctahedral shuffle complexes (theorem 8.4).
scientific article; zbMATH DE number 745284