Return to Answer

Let me work over $\mathbb{C}$ for simplicity. We have $L(V) \cong V^{\otimes n} \otimes_{S_n} \text{Lie}(n)$ where $\text{Lie}(n)$ is the $n^{th}$ space of the Lie operad; this is, to be clear, true for any $V$, in great generality, and is a special case of a very general fact about free algebras over operads $O$, namely that they are given by

$$V \mapsto \bigoplus_{n \ge 0} V^{\otimes n} \otimes_{S_n} O(n).$$

This is Proposition 5.2.5 in Loday and Vallette's Algebraic Operads; unfortunately there it is only stated for vector spaces but the argument generalizes with no modifications to $\mathbb{Z}_2$-graded vector spaces, $\mathbb{Z}$-graded vector spaces, or for that matter any cocomplete symmetric monoidal category (which, to be safe, includes the condition that tensor product is cocontinuous in each variable).

$\text{Lie}(n)$ is known to be isomorphic as an $S_n$-representation to the induced representation $\text{Ind}_{C_n}^{S_n} \chi$ where $\chi$ is the $1$-dimensional representation $k \mapsto e^{ \frac{ 2 \pi i k}{n} }$ (this is equivalent to Klyachko's theorem but stating it this way makes it clear that we don't need $V$ to be purely even). This gives

$$\begin{eqnarray*} L(V) &\cong& V^{\otimes n} \otimes_{S_n} \text{Ind}_{C^n}^{S_n} \chi \\ &\cong& V^{\otimes n} \otimes_{C_n} \chi \end{eqnarray*}.$$

We, again, do not need $V$ to be purely even here; if $V$ has an odd component then the action of $C_n$ on $V^{\otimes n}$ just has some different Koszul signs to it. Explicitly, if $v_1 \otimes \dots \otimes v_n \in V^{\otimes n}$ and each $v_i$ is homogeneous of degree $|v_i|$ then the cyclic permutation which commutes $v_1$ past all the other vectors produces

$$(-1)^{|v_1|(|v_2| + \dots + |v_n|} v_2 \otimes \dots \otimes v_n \otimes v_1.$$

For example if the $v_i$ are all odd then the sign is $(-1)^n$ which matches the sign of the $n$-cycle as a permutation in $S_n$.

I guess we can be more specific as follows. If $V$ is purely even then Schur-Weyl duality gives

$$V^{\otimes n} \cong \bigoplus_{\lambda \vdash n} S_{\lambda}(V) \otimes M^{\lambda}$$

where $S_{\lambda}$ is a Schur functor and $M^{\lambda}$ is a Specht module. This gives that $V^{\otimes n} \otimes_{C_n} \chi$ is a sum of Schur functors where $S_{\lambda}$ appears with multiplicity $\dim M^{\lambda} \otimes_{C_n} \chi$. If $V$ is purely odd then $V^{\otimes n}$ is modified as an $S_n$-representation by the sign representation which has the effect of replacing every $M^{\lambda}$ with $M^{\lambda^T}$ where $\lambda^T$ is the conjugate partition (I forget what the standard notation for this is), or equivalently replacing every $S_{\lambda}$ with $S_{\lambda^T}$. And the general case is a free product of the purely even and purely odd cases.

Let me work over $\mathbb{C}$ for simplicity. We have

$$L(V) \cong \bigoplus_{n\ge0} V^{\otimes n} \otimes_{S_n} \text{Lie}(n),$$

where $\text{Lie}(n)$ is the $n^{th}$ space of the Lie operad, with specific $S_n$ actions on $\text{Lie}(n)$ and $V^{\otimes n}$ to be precised below; this is, to be clear, true for any $V$, in great generality, and is a special case of a very general fact about free algebras over symmetric operads $O$, namely that they are given by

$$V \mapsto \bigoplus_{n \ge 0} V^{\otimes n} \otimes_{S_n} O(n).$$

This is Proposition 5.2.5 in Loday and Vallette's Algebraic Operads; unfortunately there it is only stated for vector spaces but the argument generalizes with no modifications to $\mathbb{Z}_2$-graded vector spaces, $\mathbb{Z}$-graded vector spaces, or for that matter any cocomplete symmetric monoidal category (which, to be safe, includes the condition that tensor product is cocontinuous in each variable).

In the graded case, the action of $S_n$ on $V^{\otimes n}$ is by permutations with the Koszul sign, namely it is generated by the action

$$v_1 \otimes \cdots v_i \cdots v_j \cdots \otimes v_n \mapsto (-)^{|v_i||v_j|} v_1 \otimes \cdots v_j \cdots v_i \cdots \otimes v_n$$

of permutations $(i j)$ on products of homogeneous elements. Then, $\text{Lie}(n)$ is known to be isomorphic as an $S_n$-representation to the induced representation $\text{Ind}_{C_n}^{S_n} \chi$, where $C_n$ is the cyclic group of order $n$ generated by the cyclic permutation $c = (12\cdots n)$ and $\chi$ is its $1$-dimensional representation $c^k \mapsto e^{ \frac{ 2 \pi i k}{n} }$. This gives

$$\begin{eqnarray*} L^n(V) &\cong& V^{\otimes n} \otimes_{S_n} \text{Ind}_{C^n}^{S_n} \chi \\ &\cong& V^{\otimes n} \otimes_{C_n} \chi \end{eqnarray*}.$$

This is equivalent to Klyachko's theorem in the purely even case, but stating it this way makes it clear that the graded case is taken care of by the action of $S_n$ on $V^{\otimes n}$. Explicitly, if $v_1 \otimes \dots \otimes v_n \in V^{\otimes n}$ and each $v_i$ is homogeneous of degree $|v_i|$ then the cyclic permutation which commutes $v_1$ past all the other vectors produces

$$(-1)^{|v_1|(|v_2| + \dots + |v_n|)} v_2 \otimes \dots \otimes v_n \otimes v_1.$$

For example if the $v_i$ are all odd then the sign is $(-1)^n$ which matches the sign of the $n$-cycle as a permutation in $S_n$.

I guess we can be even more specific as follows. If $V$ is purely even then Schur-Weyl duality gives

$$V^{\otimes n} \cong \bigoplus_{\lambda \vdash n} S_{\lambda}(V) \otimes M^{\lambda}$$

where $S_{\lambda}$ is a Schur functor and $M^{\lambda}$ is a Specht module. This gives that $V^{\otimes n} \otimes_{C_n} \chi$ is a sum of Schur functors where $S_{\lambda}$ appears with multiplicity $\dim M^{\lambda} \otimes_{C_n} \chi$. If $V$ is purely odd then $V^{\otimes n}$ is modified as an $S_n$-representation by the sign representation which has the effect of replacing every $M^{\lambda}$ with $M^{\lambda^T}$ where $\lambda^T$ is the conjugate partition (I forget what the standard notation for this is), or equivalently replacing every $S_{\lambda}$ with $S_{\lambda^T}$. And the general case is a free product of the purely even and purely odd cases.

Let me work over $\mathbb{C}$ for simplicity. We have $L(V) \cong V^{\otimes n} \otimes_{S_n} \text{Lie}(n)$ where $\text{Lie}(n)$ is the $n^{th}$ space of the Lie operad; this is, to be clear, true for any $V$, in great generality, and is a special case of a very general fact about free algebras over operads $O$, namely that they are given by

$$V \mapsto \bigoplus_{n \ge 0} V^{\otimes n} \otimes_{S_n} O(n).$$

This is Proposition 5.2.5 in Loday and Vallette's Algebraic Operads; unfortunately there it is only stated for vector spaces but the argument generalizes with no modifications to $\mathbb{Z}_2$-graded vector spaces, $\mathbb{Z}$-vector spaces, or for that matter any cocomplete symmetric monoidal category (which, to be safe, includes the condition that tensor product is cocontinuous in each variable).

$\text{Lie}(n)$ is known to be isomorphic as an $S_n$-representation to the induced representation $\text{Ind}_{C_n}^{S_n} \chi$ where $\chi$ is the $1$-dimensional representation $k \mapsto e^{ \frac{ 2 \pi i k}{n} }$ (this is equivalent to Klyachko's theorem but stating it this way makes it clear that we don't need $V$ to be purely even). This gives

$$\begin{eqnarray*} L(V) &\cong& V^{\otimes n} \otimes_{S_n} \text{Ind}_{C^n}^{S_n} \chi \\ &\cong& V^{\otimes n} \otimes_{C_n} \chi \end{eqnarray*}.$$

We, again, do not need $V$ to be purely even here; if $V$ has an odd component then the action of $C_n$ on $V^{\otimes n}$ just has some different Koszul signs to it. Explicitly, if $v_1 \otimes \dots \otimes v_n \in V^{\otimes n}$ and each $v_i$ is homogeneous of degree $|v_i|$ then the cyclic permutation which commutes $v_1$ past all the other vectors produces

$$(-1)^{|v_1|(|v_2| + \dots + |v_n|} v_2 \otimes \dots \otimes v_n \otimes v_1.$$

For example if the $v_i$ are all odd then the sign is $(-1)^n$ which matches the sign of the $n$-cycle as a permutation in $S_n$.

I guess we can be more specific as follows. If $V$ is purely even then Schur-Weyl duality gives

$$V^{\otimes n} \cong \bigoplus_{\lambda \vdash n} S_{\lambda}(V) \otimes M^{\lambda}$$

where $S_{\lambda}$ is a Schur functor and $M^{\lambda}$ is a Specht module. This gives that $V^{\otimes n} \otimes_{C_n} \chi$ is a sum of Schur functors where $S_{\lambda}$ appears with multiplicity $\dim M^{\lambda} \otimes_{C_n} \chi$. If $V$ is purely odd then $V^{\otimes n}$ is modified as an $S_n$-representation by the sign representation which has the effect of replacing every $M^{\lambda}$ with $M^{\lambda^T}$ where $\lambda^T$ is the conjugate partition (I forget what the standard notation for this is), or equivalently replacing every $S_{\lambda}$ with $S_{\lambda^T}$. And the general case is a free product of the purely even and purely odd cases.

Let me work over $\mathbb{C}$ for simplicity. We have $L(V) \cong V^{\otimes n} \otimes_{S_n} \text{Lie}(n)$ where $\text{Lie}(n)$ is the $n^{th}$ space of the Lie operad; this is, to be clear, true for any $V$, in great generality, and is a special case of a very general fact about free algebras over operads $O$, namely that they are given by

$$V \mapsto \bigoplus_{n \ge 0} V^{\otimes n} \otimes_{S_n} O(n).$$

This is Proposition 5.2.5 in Loday and Vallette's Algebraic Operads; unfortunately there it is only stated for vector spaces but the argument generalizes with no modifications to $\mathbb{Z}_2$-graded vector spaces, $\mathbb{Z}$-graded vector spaces, or for that matter any cocomplete symmetric monoidal category (which, to be safe, includes the condition that tensor product is cocontinuous in each variable).

$\text{Lie}(n)$ is known to be isomorphic as an $S_n$-representation to the induced representation $\text{Ind}_{C_n}^{S_n} \chi$ where $\chi$ is the $1$-dimensional representation $k \mapsto e^{ \frac{ 2 \pi i k}{n} }$ (this is equivalent to Klyachko's theorem but stating it this way makes it clear that we don't need $V$ to be purely even). This gives

$$\begin{eqnarray*} L(V) &\cong& V^{\otimes n} \otimes_{S_n} \text{Ind}_{C^n}^{S_n} \chi \\ &\cong& V^{\otimes n} \otimes_{C_n} \chi \end{eqnarray*}.$$

We, again, do not need $V$ to be purely even here; if $V$ has an odd component then the action of $C_n$ on $V^{\otimes n}$ just has some different Koszul signs to it. Explicitly, if $v_1 \otimes \dots \otimes v_n \in V^{\otimes n}$ and each $v_i$ is homogeneous of degree $|v_i|$ then the cyclic permutation which commutes $v_1$ past all the other vectors produces

$$(-1)^{|v_1|(|v_2| + \dots + |v_n|} v_2 \otimes \dots \otimes v_n \otimes v_1.$$

For example if the $v_i$ are all odd then the sign is $(-1)^n$ which matches the sign of the $n$-cycle as a permutation in $S_n$.

I guess we can be more specific as follows. If $V$ is purely even then Schur-Weyl duality gives

$$V^{\otimes n} \cong \bigoplus_{\lambda \vdash n} S_{\lambda}(V) \otimes M^{\lambda}$$

where $S_{\lambda}$ is a Schur functor and $M^{\lambda}$ is a Specht module. This gives that $V^{\otimes n} \otimes_{C_n} \chi$ is a sum of Schur functors where $S_{\lambda}$ appears with multiplicity $\dim M^{\lambda} \otimes_{C_n} \chi$. If $V$ is purely odd then $V^{\otimes n}$ is modified as an $S_n$-representation by the sign representation which has the effect of replacing every $M^{\lambda}$ with $M^{\lambda^T}$ where $\lambda^T$ is the conjugate partition (I forget what the standard notation for this is), or equivalently replacing every $S_{\lambda}$ with $S_{\lambda^T}$. And the general case is a free product of the purely even and purely odd cases.

Let me work over $\mathbb{C}$ for simplicity. We have $L(V) \cong V^{\otimes n} \otimes_{S_n} \text{Lie}(n)$ where $\text{Lie}(n)$ is the $n^{th}$ space of the Lie operad; this is, to be clear, true for any $V$, in great generality, and is a special case of a very general fact about free algebras over operads $O$, namely that they are given by

$$V \mapsto \bigoplus_{n \ge 0} V^{\otimes n} \otimes_{S_n} O(n).$$

$\text{Lie}(n)$ is known to be isomorphic as an $S_n$-representation to the induced representation $\text{Ind}_{C_n}^{S_n} \chi$ where $\chi$ is the $1$-dimensional representation $k \mapsto e^{ \frac{ 2 \pi i k}{n} }$ (I don't have a reference for this or know who it's due to). This gives

$$\begin{eqnarray*} L(V) &\cong& V^{\otimes n} \otimes_{S_n} \text{Ind}_{C^n}^{S_n} \chi \\ &\cong& V^{\otimes n} \otimes_{C_n} \chi \end{eqnarray*}$$

which is equivalent to Klyachko's theorem when $V$ is purely even. The statement is just the same if $V$ has an odd component except that the action of $C_n$ on $V^{\otimes n}$ has some different Koszul signs to it. Explicitly, if $v_1 \otimes \dots \otimes v_n \in V^{\otimes n}$ and each $v_i$ is homogeneous of degree $|v_i|$ then the cyclic permutation which commutes $v_1$ past all the other vectors produces

$$(-1)^{|v_1|(|v_2| + \dots + |v_n|} v_2 \otimes \dots \otimes v_n \otimes v_1.$$

For example if the $v_i$ are all odd then the sign is $(-1)^n$ which matches the sign of the $n$-cycle as a permutation in $S_n$.

I guess we can be more specific as follows. If $V$ is purely even then Schur-Weyl duality gives

$$V^{\otimes n} \cong \bigoplus_{\lambda \vdash n} S_{\lambda}(V) \otimes M^{\lambda}$$

where $S_{\lambda}$ is a Schur functor and $M^{\lambda}$ is a Specht module. This gives that $V^{\otimes n} \otimes_{C_n} \chi$ is a sum of Schur functors where $S_{\lambda}$ appears with multiplicity $\dim M^{\lambda} \otimes_{C_n} \chi$. If $V$ is purely odd then $V^{\otimes n}$ is modified as an $S_n$-representation by the sign representation which has the effect of replacing every $M^{\lambda}$ with $M^{\lambda^T}$ where $\lambda^T$ is the conjugate partition (I forget what the standard notation for this is), or equivalently replacing every $S_{\lambda}$ with $S_{\lambda^T}$. And the general case is a free product of the purely even and purely odd cases.

Let me work over $\mathbb{C}$ for simplicity. We have $L(V) \cong V^{\otimes n} \otimes_{S_n} \text{Lie}(n)$ where $\text{Lie}(n)$ is the $n^{th}$ space of the Lie operad; this is, to be clear, true for any $V$, in great generality, and is a special case of a very general fact about free algebras over operads $O$, namely that they are given by

$$V \mapsto \bigoplus_{n \ge 0} V^{\otimes n} \otimes_{S_n} O(n).$$

This is Proposition 5.2.5 in Loday and Vallette's Algebraic Operads; unfortunately there it is only stated for vector spaces but the argument generalizes with no modifications to $\mathbb{Z}_2$-graded vector spaces, $\mathbb{Z}$-vector spaces, or for that matter any cocomplete symmetric monoidal category (which, to be safe, includes the condition that tensor product is cocontinuous in each variable).

$\text{Lie}(n)$ is known to be isomorphic as an $S_n$-representation to the induced representation $\text{Ind}_{C_n}^{S_n} \chi$ where $\chi$ is the $1$-dimensional representation $k \mapsto e^{ \frac{ 2 \pi i k}{n} }$ (this is equivalent to Klyachko's theorem but stating it this way makes it clear that we don't need $V$ to be purely even). This gives

$$\begin{eqnarray*} L(V) &\cong& V^{\otimes n} \otimes_{S_n} \text{Ind}_{C^n}^{S_n} \chi \\ &\cong& V^{\otimes n} \otimes_{C_n} \chi \end{eqnarray*}.$$

We, again, do not need $V$ to be purely even here; if $V$ has an odd component then the action of $C_n$ on $V^{\otimes n}$ just has some different Koszul signs to it. Explicitly, if $v_1 \otimes \dots \otimes v_n \in V^{\otimes n}$ and each $v_i$ is homogeneous of degree $|v_i|$ then the cyclic permutation which commutes $v_1$ past all the other vectors produces

$$(-1)^{|v_1|(|v_2| + \dots + |v_n|} v_2 \otimes \dots \otimes v_n \otimes v_1.$$

For example if the $v_i$ are all odd then the sign is $(-1)^n$ which matches the sign of the $n$-cycle as a permutation in $S_n$.

I guess we can be more specific as follows. If $V$ is purely even then Schur-Weyl duality gives

$$V^{\otimes n} \cong \bigoplus_{\lambda \vdash n} S_{\lambda}(V) \otimes M^{\lambda}$$

where $S_{\lambda}$ is a Schur functor and $M^{\lambda}$ is a Specht module. This gives that $V^{\otimes n} \otimes_{C_n} \chi$ is a sum of Schur functors where $S_{\lambda}$ appears with multiplicity $\dim M^{\lambda} \otimes_{C_n} \chi$. If $V$ is purely odd then $V^{\otimes n}$ is modified as an $S_n$-representation by the sign representation which has the effect of replacing every $M^{\lambda}$ with $M^{\lambda^T}$ where $\lambda^T$ is the conjugate partition (I forget what the standard notation for this is), or equivalently replacing every $S_{\lambda}$ with $S_{\lambda^T}$. And the general case is a free product of the purely even and purely odd cases.