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Added a reference to today’s
Here is something I find confusing. These are functors that can be defined on monoidal categories which are also enriched over Vect and have direct sums. But these functors are just monoidal with respect to the tensor product right? They don’t respect the direct sums? E.g. the $n$th tensor power is a Schur functor, but obviously $(X+Y)^{\otimes n} \neq X^{\otimes n}+Y^{\otimes n}$. So by functor we only mean monoidal functor but without any exactness and whatnot related to sums?
I think Schur functors are rarely strong monoidal with respect to the tensor product. For instance if $X,Y$ are two finite-dimensional vector spaces, then $\bigwedge^n(X \otimes Y)$ is of dimension $\binom{\mathrm{dim}(X)\mathrm{dim}(Y)}{n}$ whereas $\big(\bigwedge^n X\big) \otimes \big(\bigwedge^n Y\big)$ is of dimension $\binom{\mathrm{dim}(X)}{n}\binom{\mathrm{dim}(Y)}{n}$. I would be less surprised if they turned out to be lax or oplax monoidal though.
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