A modular version of Klyachko's theorem on Lie representations of the general linear group. (Q2898404)

For a vector space \(V\) over an infinite field \(K\) let \(T=T(V)\) be its tensor algebra, \(L(V)\subset T(V)\) the free Lie algebra generated by \(V\), and \(L^r(V)=L(V)\cap V^{\otimes r}\). Both \(V^{\otimes r}\) and \(L^r(V)\) are \(\mathrm{GL}(V)\)-modules in a natural way and it is known that \(L^r(V)\) is a direct summand of \(V^{\otimes r}\) provided \(p=\text{char}(K)\nmid r\). If \(p=0\), then \(V^{\otimes r}\) is totally reducible and the isomorphism types of its simple summands are identified with modules \(T(\lambda)\) labelled by partitions \(\lambda\) of \(r\) into \(n=\dim(V)\) parts. By a result of \textit{A. A. Klyachko} [Sib. Mat. Zh. 15, 1296-1304 (1974; Zbl 0315.15015); translation in Sib. Math. J. 15(1974), 914-920 (1975)], all these modules appear as summands also in \(L^r(V)\) with the exceptions of \(\lambda=(r)\) (the \(r\)-th symmetric power of \(V\)) and \(\lambda= (1^r)\) (the \(r\)-th exterior power of \(V\)), provided \(r\neq 4,6\).NEWLINENEWLINE The authors consider a modular version of Klyachko's result. Here, \(V^{\otimes r}\) is a direct sum of indecomposable modules whose isomorphism types are given by the ``tilting'' modules \(T(\lambda)\) labelled by the set \(\Lambda^+_{\text{row}}(n,r)\) of row \(p\)-regular partitions \(\lambda\) of \(r\) into at most \(n\) parts. They show that for \(r=p^mk\), \(m\geq 0\), \(k>2\), \(p\nmid k\), \(r\neq 4,6\), such \(T(\lambda)\) is isomorphic to a direct summand of \(L^r(V)\) with the exceptions of \(\lambda =(r)\) and \(\lambda=\nu\), where \(\nu\) is a partition of \(r\) with transpose \(\nu'\) of the form \(\nu'=(p-1,\ldots,p-1,b)\), \(1\leq b\leq p-1\), identifying also the additional exceptions that appear when \(r=4,6\).NEWLINENEWLINE The proof is quite technical and uses the identification of certain direct summands of \(L^r(V)\) which are also direct summands of \(V^{\otimes r}\) obtained by \textit{R. M. Bryant} and \textit{M. Schocker}, [in Proc. Lond. Math. Soc. (3) 93, No. 1, 175-196 (2006; Zbl 1174.17006)]. -- In Section 2, however, some combinatorial results on Lyndon words in a lexicographically ordered free monoid are presented that may be of independent interest.