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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 3rd 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded a reference to today's * [[John Baez]], [[Joe Moeller]], [[Todd Trimble]], _Schur functors and categorified plethysm_, ([arXiv:2106.00190](https://arxiv.org/abs/2106.00190)) <a href="https://ncatlab.org/nlab/revision/diff/Schur+functor/95">diff</a>, <a href="https://ncatlab.org/nlab/revision/Schur+functor/95">v95</a>, <a href="https://ncatlab.org/nlab/show/Schur+functor">current</a>

   Added a reference to today’s

   • John Baez, Joe Moeller, Todd Trimble, Schur functors and categorified plethysm, (arXiv:2106.00190)

   diff, v95, current

2. • CommentRowNumber2.
   • CommentAuthorGuest
   • CommentTimeJul 14th 2022
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   Author: Guest
   Format: TextI added the 'local naturality' requirement to the definition of a pseudonatural transformation.

   I added the 'local naturality' requirement to the definition of a pseudonatural transformation.
3. • CommentRowNumber3.
   • CommentAuthorperezl.alonso
   • CommentTimeAug 23rd 2024
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   Author: perezl.alonso
   Format: MarkdownItexHere 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?

   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?

4. • CommentRowNumber4.
   • CommentAuthorJ-B Vienney
   • CommentTimeAug 23rd 2024
   • (edited Aug 23rd 2024)
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   Author: J-B Vienney
   Format: MarkdownItexI 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.

   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.