Timeline for Maps between general linear group that can be extended to functor
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 31, 2013 at 2:38 | comment | added | S. Carnahan♦ | Thanks. I was a bit careless. (I should try writing out the details at some point - I haven't thought about these things in a while) | |
| Oct 31, 2013 at 2:37 | history | edited | S. Carnahan♦ | CC BY-SA 3.0 |
Minor repair
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| Oct 30, 2013 at 1:42 | comment | added | ECortez | Also, I think so, that it should be endofunctors, and additive endofunctors are just $M_n(k)$s. I am wondering there is any other notion of naturallity, that containing things like tensor product. Nevertheless, thank everyone for indicating Schur functors. | |
| Oct 30, 2013 at 1:36 | comment | added | ECortez | Thanks for your answer, Carnahan. My question was not well-posed, and your interpretation seems to be good. | |
| Oct 28, 2013 at 18:53 | comment | added | Qiaochu Yuan | For a simple example, complex conjugation gives an endofunctor of finite-dimensional complex vector spaces which is not naturally isomorphic to the identity (the only Schur functor which it could possibly be), e.g. by taking traces. $\mathbb{R}$ is somewhat special in that it has no nontrivial endomorphisms. | |
| Oct 28, 2013 at 17:58 | comment | added | Qiaochu Yuan | I think I believe this statement over $\mathbb{R}$ but I'm a little confused about other fields. First, weird things happen in positive characteristic involving divided powers. Second, even in characteristic zero it seems that the category of additive endofunctors is the category of representations of the underlying field $k$ into the rings $M_n(k)$ (all regarded as rings, not $k$-algebras). In order for the stated equivalence to hold every such representation must be conjugate to the diagonal embedding but I don't see how to rule out the possibility of pathological embeddings. | |
| Oct 28, 2013 at 17:52 | comment | added | Qiaochu Yuan | Just endofunctors, I think. Most of them are not additive. | |
| Oct 28, 2013 at 8:45 | history | answered | S. Carnahan♦ | CC BY-SA 3.0 |