I am trying to gain a better understanding of Schur-Weyl duality specifically applied to symmetric functions. My motivating example is trying to understand the Frobenius character of the multilinear component of the free Lie algebra (Theorem 8.1 in Reutenauer's book on the subject), but my general confusion is more than just that example. I feel comfortable with representation theory of finite groups and less so with representation theory of Lie groups.

In several sources, the statement of Schur-Weyl duality is framed as the $\operatorname{GL}(V)\times \mathfrak{S}_k$ isomorphism of the tensor algebra $V^{\otimes k}$ with the direct sum of tensor products $M^\lambda \otimes \mathbb{S}^\lambda(V)$ of Specht modules with Schur functors indexed by partitions of $k$ into at most $\dim(V)$-many parts. As I understand it, one can then determine the multiplicity of either a Schur functor (by looking at the dimension of the corresponding Specht module) or the multiplicity of a Specht module (by looking at the dimension of the corresponding Schur functor).

I also seem to understand that, as explained in the second appendix of Stanley's Enumerative Combinatorics Vol. 2, the character value of $A \in \operatorname{GL}(V)$ for an irreducible rational $\operatorname{GL}(V)$ representation $\mathbb{S}^\lambda$ is given by evaluating the Schur polynomial on the eigenvalues of $A$.

Reutenauer's proof of theorem 8.1 (that two particular symmetric functions are equal) is as follows:

This is the Schur-Weyl duality between the representations of the symmetric group and the general linear group (Weyl 1946, Theorem 7.6.F; Macdonald 1979, A7 in Chapter 1).

My first confusion is that the references I find refer to Schur-Weyl duality as a statement about the tensor algebra, not any other algebra. The universal enveloping algebra of the free Lie algebra can be constructed as a quotient of the tensor algebra, so I'm willing to suspend some disbelief that the statement is still applicable here.

Even with that, I am having trouble connecting the dots here. Reutenauer sets up a $\operatorname{End}(V) \times \mathfrak{S}_k$-representation, and then looks at the action of a copy of $\mathfrak{S}_k$ living inside of $\operatorname{End}(V)$ and considers the Frobenius characteristic of the $\mathfrak{S}_k$-representation found by restricting the action of $\operatorname{End}(V)$. He then compares this to the multivariate generating function of dimensions of homogeneous polynomials of multidegree $\alpha$ which are stable under the $\mathfrak{S}_k \subseteq \operatorname{End}(V)$ action and asserts they are the same via the proof quoted above.

The character value of the restriction of a representation is just the character value of the original representation. The eigenvalues of any $A \in \mathfrak{S} \leq \operatorname{End}(V)$ should be all $1$. Then I think the character value of $A$ should be the number of semistandard Young tableaux of shape $\lambda$ with entries at most $\dim(V)$ (i.e. $s_\lambda(1,1,\ldots,1)$). But I am struggling to see why this proves Reutenauer's claim.

Any help or general information about the relationship between Schur-Weyl duality and symmetric functions you could provide would be greatly appreciated.

Edit: I made some mistakes in my question including:

• The universal enveloping algebra is the tensor algebra, not a (nontrivial) quotient.
• Eigenvalues of permutation matrices are roots of unity, not 1.