Lie powers of the natural module for GL(2,K)

In recent work of R. M. Bryant and the second author a (partial) modular analogue of Klyachko's 1974 result on Lie powers of the natural $\rm{GL}(n,K)$ was presented. There is was shown that nearly all of the indecomposable summands of the $r$th tensor power also occur up to isomorphism as summands of the $r$th Lie power provided that $r\neq p^m$ and $r \neq 2p^m$, where $p$ is the characteristic of $K$. In the current paper we restrict attention to ${\rm GL}(2,K)$ and consider the missing cases where $r = p^m$ and $r = 2p^m$. In particular, we prove that the indecomposable summand of the $r$th tensor power of the natural module with highest weight $(r-1,1)$ is a summand of the $r$th Lie power if and only if $r$ is a not power of $p$

Products of homogeneous subspaces in free Lie algebras

Let $L$ be a free Lie algebra of rank $k$ over a field $K$. If $n\geq 1$ then we let $L_n$ denote the homogeneous subspace of $L$ spanned by Lie products of weight $n$ in the free generators of $L$. We obtain formulae for the dimensions of the subspaces $[L_m,L_n]$ for all $m$ and $n$. This answers a question raised by Eamonn O'Brien in private communication

ON THE DIMENSION OF PRODUCTS OF HOMOGENEOUS SUBSPACES IN FREE LIE ALGEBRAS

Let $L$ be a free Lie algebra of finite rank over a field $K$ and let $L_{n}$ denote the degree $n$ homogeneous component of $L$. Formulae for the dimension of the subspaces $[L_m,L_n]$ for all $m$ and $n$ were obtained by the second author and Michael Vaughan-Lee. In this note we consider subspaces of the form $[L_m,L_n,L_k]=[[L_m,L_n],L_k]$. Surprisingly, in contrast to the case of a product of two homogeneous components, the dimension of such products may depend on the characteristic of the field $K$. For example, the dimension of $[L_2,L_2,L_1]$ over fields of characteristic $2$ is different from the dimension over fields of characteristic other than $2$. Our main result are formulae for the dimension of $[L_m,L_n,L_k]$. Under certain conditions on $m$, $n$ and $k$ they lead to explicit formulae that do not depend on the characteristic of $K$, and express the dimension of $[L_m,L_n,L_k]$ in terms of Witt's dimension function