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Stream: learning: questions

Topic: monomial endofunctors

Jean-Baptiste Vienney (Apr 21 2022 at 03:02):

Let $n \ge 0$ . I'm looking for a nice class of endofunctors on the category of k-finite-dimensional vector spaces, k of characteristic $0$ , which contains the tensor power $\_^{\otimes n}$ , the symmetric power $\_^{\otimes_{s}n}$ , the exterior power $\bigwedge^{n}$ . I know that the concepts of Schur functors and Polynomial functors exists and thus we can surely anwser the question with them, but I'm not very familiar with them.

In positive characteristic, I imagine than we should add the divided power $\Gamma^{n}$ in the list.

John Baez (Apr 21 2022 at 04:12):

For characteristic zero you're talking about Schur functors. I wrote a paper about them:

John Baez, Joe Moeller and Todd Trimble, Schur functors and categorified plethysm.

The first half of this paper provides a modern outlook on Schur functors based on category theory. But it may be a bit excess to requirements if you're only trying to understand the basics of Schur functors.

John Baez (Apr 21 2022 at 04:13):

For positive characteristic the story will change, as you say, and I'm very interested in that, but I haven't investigated it along the lines that our paper would suggest.

Simon Burton (Apr 21 2022 at 11:29):

How does the divided power operator work ?

John Baez (Apr 21 2022 at 14:37):

It's an operation that for real numbers is $x^n/n!$ , but it can sometimes be defined in situations where you can't literally divide. You can axiomatize the properties these operations should have and get the concept of [[divided power algebra]].

Simon Burton (Apr 21 2022 at 16:16):

So it's another categorification of $x^n/n!$ ? As usual it would be good to see some smallest non-trivial examples..

I have spent some time fooling around with Schur functors in characteristic two, because this is where classical linear codes (& quantum codes) live.. But I got totally lost trying to come up with an exterior power that makes sense in this case.