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Let $V$ be a vector space over a field of characteristic $0$, and let $L_k(V)$ be the degree $k$ part of the free Lie algebra over $V$. There is an exact sequence $$0\to D_n(V)\to L_1(V)\otimes L_{n+1}(V)\to L_{n+2}V\to 0$$ where the map on the right is the Lie bracket and $D_n(V)$ is defined as the kernel of this map. Now $D_n(V)$ is a $GL(V)$-module, and I'm curious about how it decomposes as a direct sum of irreducibles. By playing around with dimension formulas, I was able to show that $D_1(V)=S^{(1,1,1)}V$, $D_2(V)=S^{(2,2)}V$ and $D_3(V)=S^{(3,1,1)}V$. Here $S^\lambda(V)$ is the irreducible representation of $GL(V)$ corresponding to partition $\lambda$. So in fact, if I haven't made a mistake, $D_k(V)$ is actually irreducible for $k=1,2,3$, with nice symmetric looking partitions.

Is $D_k(V)$ always irreducible, and is there an easy way to tell what the partition is? If not, does anyone know what is known about this?

If it helps, here is an alternate way of characterizing $D_n$. Consider the Lie operad, and look at the part with a total of $n+2$ input/output slots, denoted $Lie((n+2))$. Then $$D_n\cong [V^{\otimes n+2}\otimes Lie((n+2))]_{Sym(n+2)},$$ which denotes the coinvariants under the action of $Sym(n)$ acting simultaneously on $Lie((n+2))$ and $V^{\otimes n}$, in the latter case by permuting the factors. This is isomorphic to a space of planar unitrivalent trees with leaves labeled by vectors from $V$, modulo antisymmetry and Jacobi (IHX) relations.

I don't know very much representation theory, so it's possible my question has an easy answer.

Added 9/20/2011 Morita, Sakasai and Suzuki just posted a preprint proving that $D_{4k+2}$ and $D_{4k+3}$ always have symmetric decompositions, in the sense that there is a symmetry in the corresponding Young diagrams exchanging rows and columns. This is a neat result.

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Let $V$ be a vector space over a field of characteristic $0$, and let $L_k(V)$ be the degree $k$ part of the free Lie algebra over $V$. There is an exact sequence $$0\to D_n(V)\to L_1(V)\otimes L_{n+1}(V)\to L_{n+2}V\to 0$$ where the map on the right is the Lie bracket and $D_n(V)$ is defined as the kernel of this map. Now $D_n(V)$ is a $GL(V)$-module, and I'm curious about how it decomposes as a direct sum of irreducibles. By playing around with dimension formulas, I was able to show that $D_1(V)=S^{(1,1,1)}V$, $D_2(V)=S^{(2,2)}V$ and $D_3(V)=S^{(3,1,1)}V$. Here $S^\lambda(V)$ is the irreducible representation of $GL(V)$ corresponding to partition $\lambda$. So in fact, if I haven't made a mistake, $D_k(V)$ is actually irreducible for $k=1,2,3$, with nice symmetric looking partitions.

Is $D_k(V)$ always irreducible, and is there an easy way to tell what the partition is? If not, does anyone know what is known about this?

If it helps, here is an alternate way of characterizing $D_n$. Consider the Lie operad, and look at the part with a total of $n+2$ input/output slots, denoted $Lie((n))$. Lie((n+2))$. Then $$D_n\cong [V^{\otimes n+2}\otimes Lie((n+2))]_{Sym(n+2)},$$ which denotes the coinvariants under the action of$Sym(n)$acting simultaneously on$Lie((n+2))$and$V^{\otimes n}$, in the latter case by permuting the factors. This is isomorphic to a space of planar unitrivalent trees with leaves labeled by vectors from$V$, modulo antisymmetry and Jacobi (IHX) relations. I don't know very much representation theory, so it's possible my question has an easy answer. 1 # GL(V)-representation theory for a Lie bracket kernel Let$V$be a vector space over a field of characteristic$0$, and let$L_k(V)$be the degree$k$part of the free Lie algebra over$V$. There is an exact sequence $$0\to D_n(V)\to L_1(V)\otimes L_{n+1}(V)\to L_{n+2}V\to 0$$ where the map on the right is the Lie bracket and$D_n(V)$is defined as the kernel of this map. Now$D_n(V)$is a$GL(V)$-module, and I'm curious about how it decomposes as a direct sum of irreducibles. By playing around with dimension formulas, I was able to show that$D_1(V)=S^{(1,1,1)}V$,$D_2(V)=S^{(2,2)}V$and$D_3(V)=S^{(3,1,1)}V$. Here$S^\lambda(V)$is the irreducible representation of$GL(V)$corresponding to partition$\lambda$. So in fact, if I haven't made a mistake,$D_k(V)$is actually irreducible for$k=1,2,3$, with nice symmetric looking partitions. Is$D_k(V)$always irreducible, and is there an easy way to tell what the partition is? If not, does anyone know what is known about this? If it helps, here is an alternate way of characterizing$D_n$. Consider the Lie operad, and look at the part with a total of$n+2$input/output slots, denoted$Lie((n))$. Then $$D_n\cong [V^{\otimes n+2}\otimes Lie((n+2))]_{Sym(n+2)},$$ which denotes the coinvariants under the action of$Sym(n)$acting simultaneously on$Lie((n+2))$and$V^{\otimes n}$, in the latter case by permuting the factors. This is isomorphic to a space of planar unitrivalent trees with leaves labeled by vectors from$V\$, modulo antisymmetry and Jacobi (IHX) relations.

I don't know very much representation theory, so it's possible my question has an easy answer.