Some calculations of \(\mathrm{Lie}(n)^{\max}\) for low \(n\)
Let \(V\) be an \(n\)-dimensional vector space over a field \(k\). Then \(\text{Lie}(n)\) is the linear span within the \(n\)-fold tensor product of \(V\) of all \(n\)-fold iterated commutators in which each basis element of \(V\) appears (exactly) once. This is naturally a module over the group ring of the \(n\)-th symmetric group. This module appears in important work of topologists related to the homology of configuration spaces, cf. \textit{F. R. Cohen} [Contemp. Math. 188, 49--55 (1995; Zbl 0849.55015)], and to Goodwillie towers, cf. \textit{G. Arone} and \textit{M. Mahowald} [Invent. Math. 135, No. 3, 743--788 (1999; Zbl 0997.55016)]. The object of this paper is to understand the maximum projective submodule \(\text{Lie}(n)^{max}\) of \(\text{Lie}(n)\) in some cases. The driving motivation here is to understand the factor \(A(V)^{min}\) which appears in a natural coalgebra decomposition of the tensor algebra \(T(V)\) and which was constructed and studied by the authors in \textit{P. Selick} and \textit{J. Wu} [Mem. Am. Math. Soc. 701 (2000; Zbl 0964.55012)]. The factor \(A(V)^{min}\) is the minimum natural coalgebra retract of \(T(V)\) which contains \(V\), and its study depends heavily on properties of the module \(\text{Lie}(n)\) and its submodule \(\text{Lie}(n)^{max}\). Note that when \(n\) is prime to the characteristic of \(k\) then \(\text{Lie}(n)\) is itself projective and \(\text{Lie}(n)^{max}=\text{Lie}(n)\). The focus of this paper is the calculation of \(\text{Lie}(n)^{max}\) for \(n < 10\) and char\((k)=2\). Since \(\text{Lie}(n)^{max}=0\) for \(n=2,4\) (\S1) or when \(n\) is odd (see above), the interesting and difficult task is to understand \(\text{Lie}(6)^{max}\) and \(\text{Lie}(8)^{max}\). Their respective dimensions are found to be \(96\) and \(4224\). This dimension count is just the start of extensive and laborious calculations where the modules in question are decomposed into explicit indecomposable projective submodules; their dimensions determined and their ``characteristic polynomials'' (\S3) computed. The techniques borrow heavily from the representation theory of the symmetric group.
scientific article; zbMATH DE number 5321431
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