In the paper `Schur functors and Schur complexes' by Akin et al., the notion of a Schur functor had been defined for the first time over an arbitrary commutative ring $R$.
To recall the definition (I use Efimov's `Derived categories of Grassmannians over integers and modular representation theory' as a reference), for a finitely generated projective module $V$ over a commutative ring $R$ and a partition $\lambda = (\lambda_{1},\dots,\lambda_{n})$ of degree $d$, the Schur functor $S_{\lambda}$ is defined as
$S_{\lambda}(V) = \text{Im}(\Lambda^{\lambda'}(V) \rightarrow V^{\otimes d} \xrightarrow{s_{\lambda}} V^{\otimes d} \rightarrow Sym^{\lambda}(V))$.
Here, $\lambda'$ is the conjugate partition of $\lambda$; $\Lambda^{\lambda}(V) := \Lambda^{\lambda_{1}}(V)\otimes\cdots\otimes \Lambda^{\lambda_{n}}(V)$; $Sym^{\lambda}(V) :=Sym^{\lambda_{1}}(V)\otimes\cdots\otimes Sym^{\lambda_{n}}(V)$; $\Lambda^{\lambda'}(V) \rightarrow V^{\otimes d}$ and $V^{\otimes d} \rightarrow Sym^{\lambda}(V)$ are the canonical inclusions and projections respectively; and $V^{\otimes d} \xrightarrow{s_{\lambda}} V^{\otimes d}$ is defined via $s_{\lambda}(v_{1}\otimes\cdots\otimes v_{d}):= v_{\sigma_{\lambda}(1)}\otimes\cdots\otimes v_{\sigma_{\lambda}(d)}$, where $\sigma_{\lambda} \in S_{d}$ is the permutation associated to $\lambda$. (For more details, Efimov has a readable description in his paper).
Now, one can take the `dual' of the above construction, which gives you the notion of a Weyl functor (this is what Akin et al. call coSchur functors).
Formally, we have the Weyl functor $W_{\lambda}$ defined as
$W_{\lambda}(V) = \text{Im}(\Gamma^{\lambda}(V) \rightarrow V^{\otimes d} \xrightarrow{s_{\lambda'}} V^{\otimes d} \rightarrow \Lambda^{\lambda'}(V))$,
where $\Gamma^{\lambda}(V) := \Gamma^{\lambda_{1}}(V) \otimes \cdots \otimes \Gamma^{\lambda_{n}}(V)$, the tensor product of the divided powers of $V$.
Note that $S_{\lambda}(V)^{*} \cong W_{\lambda}(V^{*})$.
My question is: if $R$ is assumed to be a field of characteristic zero, then $S_{\lambda} = W_{\lambda}$ (up to natural isomorphism). Why is this true?
It is not at all obvious to me (at least from the definitions) why this should be the case. Unfortunately, my knowledge of Representation theory is not very advanced, and my feeling is that this follows trivially from a well-known Representation theory fact, which is perhaps why others don't bother clarifying this.
I became interested in Schur functors as I want to understand the semi-orthogonal decomposition on the bounded derived category of Grassmannians, as given in Efimov's paper.
Any help would be much appreciated!
