### Schur Functors

#### Posted by John Baez

As part of the Tale of Groupoidification, I’ll need to talk about Schur functors. As usually defined, these are simply functors

$F: Vect_{\mathbb{C}} \to Vect_{\mathbb{C}}$

where $Vect_{\mathbb{C}}$ is the category of finite-dimensional complex vector spaces.

An example of a Schur functor is ‘take the antisymmetrized 3rd tensor power’. In the category of Schur functors, $hom(Vect,Vect)$, every object can be expressed as a direct sum of certain ‘irreducible’ objects, which correspond to Young diagrams. The example I just mentioned corresponds to this Young diagram:

Given any group representation

$R: G \to Vect_{\mathbb{C}}$

we can compose it with any Schur functor

$F: Vect_{\mathbb{C}} \to Vect_{\mathbb{C}}$

and get a new representation

$F R : G \to Vect_{\mathbb{C}}$

This is a great method of getting new reps from old.

There’s much more to say… but first, Allen Knutson has a question!

Allen asks:

For as long as I’ve understood Schur functors, I’ve thought about them as functors $Vect_{\mathbb{C}} \to Vect_{\mathbb{C}}$. But now that we’re going through them in a reading course on Fulton’s

Young Tableaux, I discover that the input isn’t really a complex vector space, but an arbitrary module over a commutative ring. (And maybe, just maybe, a bimodule over a noncommutative one, but I doubt it.) In particular, the Schur functor commutes with base change AKA extension of scalars.What is the right way to describe this object, categorically? (Or should I say, 2- or 3-categorically?)

Here’s a first stab at answering this.

Presumably ‘commuting with base change’ means that our Schur functor is not merely a functor

$F : Mod_R \to Mod_R$

from a single category of $R$-modules to itself, but a *family* of functors, one for each commutative ring $R$, depending *naturally* on $R$. So, we have some 2-category $Mod$ of ‘module categories’, and our Schur functor gives some (pseudo)natural transformation

$F : 1_{Mod} \Rightarrow 1_{Mod}$

Thus, for each object $Mod_R$ in $Mod$, we get

$F_R : Mod_R \to Mod_R$

and this is natural in some suitably weak sense.

Working out the details here would be a lot of fun:

- I’m a bit worried about what happens in nonzero characteristic. Do things like symmetrizing and antisymmetrizing really work the same when you can’t divide by some prime $p$? Or do you get different sorts of ‘Schur functors’ in nonzero characteristic?
- What really matters is not so much ‘module categories’ but symmetric monoidal abelian categories, perhaps enriched over $\Vect_{\mathbb{Q}}$ or something — whatever is the minimal context where you can do stuff like ‘symmetrize’ or ‘antisymmetrize’ tensor powers of objects.

## Re: Schur Functors

I’m a bit worried about what happens in nonzero characteristic. Do things like symmetrizing and antisymmetrizing really work the same when you can’t divide by some prime p? Or do you get different sorts of ‘Schur functors’ in nonzero characteristic?You never do divide by p. In the most basic examples, Sym

^{2}resp. Alt^{2}, you tensor the tensor square andmod outby {a⊗ b-b⊗ a} resp. by {a⊗a}, for all a,b in the module.The more general definition, on pp. 104-107, is also as a quotient, of a big tensor product by a submodule generated by a bunch of things with leading coefficient 1. Perhaps they’re a module Grobner basis and so those coefficients 1 are the only ones you need worry about. (I don’t know this to be true.)

The characteristic p issues I understand to arise here are that you can’t argue that the resulting modules have unique “highest” weights any more, with which to prove irreducibility, now that weights wrap around.