Embeddings of higher Lie modules.
Consider the general linear group \(\text{GL}(V)\) of a finite dimensional \(K\)-vector space \(V\) where \(K\) is a field of characteristic zero. The tensor algebra \(T(V)\) has the structure of both a Lie algebra and a \(\text{GL}(V)\)-module. For each positive integer \(n\), the homogeneous component \(T_n(V)\) is a \(\text{GL}(V)\)-module. Of interest here are certain \(\text{GL}(V)\)-submodules of \(T_n(V)\): the classical Lie module \(L_n(V)\) and higher Lie modules corresponding to partitions of \(n\). The Lie module \(L_n(V)=T_n(V)\cap L(V)\) where \(L(V)\) is the Lie subalgebra generated by \(V\). The higher Lie modules are defined in terms of the classical Lie modules and each \(T_n(V)\) decomposes as sum of higher Lie modules. Using idempotents in the group algebra of a symmetric group and a product formula for these idempotents found by \textit{D. Blessenohl} and \textit{H. Laue} [J. Algebra 180, No. 3, 698-724 (1996; Zbl 0864.20007)], the author shows that for \(n\geq 4\), there is a chain of embeddings of higher Lie modules for those corresponding to hook partitions. This chain of embeddings is then used to study the classical Lie modules. Each \(L_n(V)\) is the direct sum of simple modules of the form \(M_p\otimes_{KS_n}T_n(V)\) over partitions \(p\vdash n\). Here \(S_n\) is the symmetric group on \(n\) objects and \(M_p\) is the simple \(KS_n\)-module corresponding to the partition \(p\). Each \(M_p\) may appear multiple times, and the fundamental question is to determine precisely the multiplicity for each partition \(p\). For \(n\geq 3\), the author determines those partitions \(p\vdash n\) for which \(M_p\) has multiplicty at least one, thus recovering a result of \textit{A. A. Klyachko} [Sib. Math. J. 15, 914-920 (1975); translation from Sib. Mat. Zh. 15, 1296-1304 (1974; Zbl 0315.15015)]. Further, the author determines those partitions whose corresponding simple modules have multiplicity exactly one. The author ends the paper with a discussion of some relations to combinatorial data on a standard tableaux.
scientific article; zbMATH DE number 2005932