Lie powers and Witt vectors
type: Item
instance of: scholarly article
title: Lie powers and Witt vectors
zbMATH Open document ID: 1198.17004
author: Marianne Johnson
author: Roger M. Bryant
published in: Journal of Algebraic Combinatorics
publication date: 2009-06-17T00:00:00Z
review text: Let \(k\) be a field of characteristic \(p>0\) and \(G\) a group. Suppose that \(G\) has a finite dimensional representation in a space \(V\). Denote by \(L(V)\) the free Lie algebra on \(V\). Then each homogeneous component \(L^n(V)\) is a finite dimensional representation of \(G\). Let \(m\) be a positive integer which is not divisible by \(p\). As it is shown by the author and \textit{M. Schocker} in the paper [Proc. Lond. Math. Soc. (3) 93, No. 1, 175--196 (2006; Zbl 1174.17006)], there exists a \(kG\)-module \(B_{p^lm} \subseteq L^{p^lm}\) which is a direct summand of \(V^{\otimes p^lm}\) such that \[ L^{p^lm} = L^{p^l}\left(B_m\right) \oplus L^{p^l-1}\left(B_{pm}\right) \oplus \cdots \oplus L^1\left(B_{p^lm}\right). \] In the present paper it is shown that the module \(V^{p^lm}\) has a decomposition \(\bigoplus _{d\mid m}\phi(d) U_{m,d}\) for some specific modules \(U_{m,d}\) and \[ \left(V^{\otimes m}\right)^{\otimes p^lm} = \bigoplus_{\lambda \in\Lambda} U_{m,\lambda(1)}\otimes \cdots \otimes U_{m,\lambda(p^l)} \] where \(\Lambda\) is a finite set of \(p^l\)-tuples \(\left(\lambda(1), \ldots, \lambda(p^l)\right)\) with entries \(\lambda(1),\ldots \) which are divisors of \(m\). Moreover \(B_{p^lm}\) has a decomposition \[ B_{p^lm} = \bigoplus_{\lambda\in \Lambda_0} U_{m,\lambda(1)}\otimes \cdots \otimes U_{m,\lambda(p^l)} \] for some specific subset \(\Lambda_0\) in \(\Lambda\).
reviewed by: Vyacheslav A. Artamonov
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 13F35
zbMATH DE Number: 5566028
zbMATH Keywords: free Lie algebras
MaRDI profile type: Publication
full work available at URL: s10801-007-0117-9
OpenAlex ID: W2095593821
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DOI: 10.1007/S10801-007-0117-9
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description: scientific article; zbMATH DE number 5566028