Let $\mathfrak{gl}(n)$ be the general linear Lie algebra of rank $n$, and $\mathfrak{S}_d$ be the symmetric group of rank $d$. It is well-known that the Schur-Weyl duality provide a equivalence between $\mathfrak{S}_d$-mod and a subcategory of category of finite-dimensional modules over $\mathfrak{gl}(n)$.

Furthermore, for a given partition $\lambda$ of $d$, Weyl character formula says that the character of the simple $\mathfrak{gl}(n)$-module $V(\lambda)$ of highest weight is Schur function $s_{\lambda}$, while the Frobenius character formula says that the Frobenius characteristic map of the Specht module $S^{\lambda}$ is the Schur function $s_{\lambda}$ as well.

$\textbf{My Question:}$ With Weyl character formula and Frobenius character formula, how can derive one from the other? Thanks very much in advance!


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