Hyperoctahedral operations on Hochschild homology
type: Item
instance of: scholarly article
title: Hyperoctahedral operations on Hochschild homology
zbMATH Open document ID: 0833.16006
author: Nantel Bergeron
published in: Advances in Mathematics
publication date: 1995-05-18T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 745284
zbMATH Keywords: idempotents
zbMATH Keywords: group algebras
zbMATH Keywords: hyperoctahedral groups
zbMATH Keywords: symmetric powers
zbMATH Keywords: free Lie algebras
zbMATH Keywords: induced characters
zbMATH Keywords: Hochschild homology
zbMATH Keywords: shuffle complexes
reviewed by: Frédéric Patras
MaRDI profile type: Publication
full work available at URL: aima.1995.1011
OpenAlex ID: W1999126380
DOI: 10.1006/AIMA.1995.1011
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description: scientific article; zbMATH DE number 745284