## October 25, 2022

### Booleans, Natural Numbers, Young Diagrams, Schur Functors

#### Posted by John Baez There’s an adjunction between commutative monoids and pointed sets, which gives a comonad. Then:

Take the booleans, apply the comonad and get the natural numbers.

Take the natural numbers, apply the comonad and get Young diagrams.

Take the Young diagrams, apply the comonad and get Schur functors.

Let me explain how this works!

There’s an adjunction between commutative monoids and pointed sets. Any commutative monoid $(M,+,0)$ has an underlying pointed set $(M,0)$, so we get a functor

$U : \mathsf{CommMon} \to \mathsf{Set}_\ast$

from commutative monoids to pointed sets. And this has a left adjoint

$F: \mathsf{Set}_\ast \to \mathsf{CommMon}$

This sends any pointed set $(S,\ast)$ to the free commutative monoid on $S$ modulo the congruence relation that forces $\ast$ to be the identity. And that’s naturally isomorphic to the free commutative monoid on the set $S - \{\ast\}$.

$F U : \mathsf{CommMon} \to \mathsf{CommMon}$

My favorite 2-element commutative monoid is the booleans $B = \{0,1\}$ made into a commutative monoid using ‘or’. Its identity element is $0$.

If we take $(B, or, 0)$ and apply the functor

$U : \mathsf{CommMon} \to \mathsf{Set}_\ast$

we get the 2-element pointed set $(B,0)$. When we apply the functor

$F: \mathsf{Set}_\ast \to \mathsf{CommMon}$

to this 2-element pointed set we get $\mathbb{N}$, made into a commutative monoid using addition. The reason is that $\mathbb{N}$ is also the free commutative monoid on the 1-element set $B - \{0\}$.

If we apply the functor $U$ to $(\mathbb{N}, + , 0)$ we get the pointed set $(\mathbb{N},0)$. When we apply the functor $F$ to the pointed set $(\mathbb{N},0)$ we get a commutative monoid that’s also the free commutative monoid on the set $\mathbb{N} - \{0\} = \{1,2,3,\dots\}$. This is usually called the set of Young diagrams, since a typical element looks like

$3 + 2 + 2 + 2 + 1$

so it can be drawn like this: (I’m counting the number of boxes in columns. We can also use the other convention, where we count the number of boxes in rows. That’s actually more common.)

Note that there is an ‘empty Young diagram’ with no boxes at all, and that’s the identity element of the free commutative monoid on $\{1,2,3,\dots\}$. But there aren’t Young diagrams with a whole bunch of 0-box columns, which is why I prefer the free commutative monoid on $\{1,2,3,\dots\}$ to the free commutative monoid on $\mathbb{N}$.

Let $(Y,+,0)$ be the commutative monoid of Young diagrams, where $0$ is the empty Young diagram — the one with no boxes at all. Applying $U$ to this we get the pointed set of Young diagrams, $(Y,0)$. Applying $F$ to that we get a commutative monoid $F(Y,0)$ that’s also the free commutative monoid on the set of nonempty Young diagrams.

And this commutative monoid $F(Y,0)$ is important in representation theory! The category $\mathsf{Schur}$ of Schur functors has Young diagrams as its simple objects. But a general object is a finite direct sum of simple objects. So, the set of isomorphism classes of Schur functors is naturally isomorphic to $F(Y,0)$. And they are isomorphic as commutative monoids, where we use direct sums of Schur functors to get a monoid structure.

So here is the slogan:

Take the booleans, apply the comonad $F U$ and get the commutative monoid of natural numbers.

Take the natural numbers, apply the comonad $F U$ and get the commutative monoid of Young diagrams.

Take the Young diagrams, apply the comonad $F U$ and get the commutative monoid of isomorphism classes of Schur functors.

Of course I want to do it again, but I’m not sure where the resulting structure shows up in math. Maybe some sort of categorified representation theory?

Posted at October 25, 2022 3:07 PM UTC

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### Theo

My favorite 2-element commutative monoid — they’re all isomorphic

Isn’t $\mathbb{Z}/2\mathbb{Z}$ a commutative monoid?

Posted by: Theo Johnson-Freyd on October 25, 2022 4:43 PM | Permalink | Reply to this

### Re: Theo

(For some reason, my computer autofilled the subject with my name. I don’t know why.)

Posted by: Theo Johnson-Freyd on October 25, 2022 4:44 PM | Permalink | Reply to this

Okay, half of them are isomorphic.

$\mathbb{Z}/2$ gives the same thing when we apply the forgetful functor to pointed sets, so I could have used that equally well.

The problem with subject headers is something we don’t really know how to fix, sorry. Look at the line “Subject” as you’re composing your comment, and make sure it says something acceptable. You can change it at that time; later the only solution is for a moderator to delete your comment.

Posted by: John Baez on October 25, 2022 5:34 PM | Permalink | Reply to this

### Hmm

This seems like waaaay too little structure to deserve referencing Schur functors. Do you have any ideas on how to involve their product, coproduct, or composition?

Posted by: Allen Knutson on October 25, 2022 10:24 PM | Permalink | Reply to this

### Re: Hmm

I haven’t tried to do that. You’re right that Schur functors have a lot more structure, but I’m not applying for grant money based on this observation, just writing a fun blog article.

I’m considering very primitive structures here, which are still important. Young diagrams naturally biject to multisets of nonzero natural numbers, isomorphism classes of Schur functors naturally biject to multisets of nonempty Young diagrams — and it’s taken me a while to figure out the right (co)monad to express the ‘nonzero’ and ‘nonempty’ clauses in a positive manner.

Posted by: John Baez on October 26, 2022 11:32 AM | Permalink | Reply to this

### Re: Hmm

You could also generate this sequence by repeatedly taking free commutative semigroups starting from $\{*\}$, and then adjoining additive identity to each.

But in any case it’s just constructing fancier names for “the” countably infinite set, so there has to be something else to carry structure.

Posted by: unekdoud on October 30, 2022 1:32 PM | Permalink | Reply to this

### Re: Hmm

As sets $B, F U B, F U F U B, \dots$ are just countably infinite sets once we reach the second step. As monoids they’re just free commutative monoids on countably many generators once we reach the third step. But they get more and more structure by virtue of all the ways we can use the natural transformations $F U \Rightarrow 1$ and $1 \Rightarrow U F$.

One famous way to package a lot of this structure is the bar construction. Namely, they give a very nice simplicial commutative monoid, which is characterized by a universal property. You can think of it as the universal way of ‘puffing up’ the booleans $B$ into a simplicial commutative monoid in which all relations have been replaced by edges, all relations-between-relations have been replaced by 2-simplices, etc.

Posted by: John Baez on November 2, 2022 12:42 PM | Permalink | Reply to this

### Re: Booleans, Natural Numbers, Young Diagrams, Schur Functors

Of course I want to do it again, but I’m not sure where the resulting structure shows up in math. Maybe some sort of categorified representation theory?

Polynomial functors? These indeed show up in 2-representation theory, being related to the work of Rouquier, Khovanov, Lauda, et al.

Posted by: Anonymous on October 26, 2022 10:00 AM | Permalink | Reply to this

### Re: Booleans, Natural Numbers, Young Diagrams, Schur Functors

Have you seen people using multisets of nontrivial Schur functors? That’s the structure I’d be hoping for.

Posted by: John Baez on October 27, 2022 9:58 AM | Permalink | Reply to this

### Re: Booleans, Natural Numbers, Young Diagrams, Schur Functors

There is a result of Macdonald that says that in characteristic zero, polynomial functors are the same as multi-sets of Schur functors. There are many applications of polynomial functors, also known as linear species, in representation theory and elsewhere.

Posted by: Anonymous on October 27, 2022 11:51 AM | Permalink | Reply to this

### Re: Booleans, Natural Numbers, Young Diagrams, Schur Functors

I think my Schur functors are arbitrary finite direct sums of what you’re calling Schur functors. That is, for me a Schur functor is not just one coming from a Young diagram, but an arbitrary finite direct sum of such. So, I believe what you’re calling “Schur functors” is what I’m calling “Young diagrams”, and what you’re calling “multi-sets of Schur functors” correspond to what I’m calling “Schur functors”.

I’m looking for appearances of the next thing after that, which you might call “multi-sets of multi-sets of Schur functors”.

Posted by: John Baez on October 27, 2022 2:30 PM | Permalink | Reply to this

### Re: Booleans, Natural Numbers, Young Diagrams, Schur Functors

My apologies, I did not really read the entirety of the original blog post, and only saw the question at the end; I think that what you call a Schur functor is in fact what I called a polynomial functor (and I think you have a joint paper in which you and your co-authors re-prove the result I attributed to Macdonald!).

Nevertheless, I think that my original comment is along the right lines: Rouquier definitely had the idea of combining 2-representations in his sense into larger structures by summing them and taking their tensor product in some sense, and since polynomial functors can be used to define 2-representations in Rouquier’s sense, one will obtain interesting mathematics of the kind you have in mind in this way. One of Rouquier’s original motivations was to construct a 4D-TQFT in this way; it quickly leads to deep mathematics. I’m not sure how far Rouquier’s programme has come, at least in terms of published material.

Posted by: Anonymous on October 27, 2022 12:23 PM | Permalink | Reply to this

### Re:

This nice post does not seem to have a title, and presumably for that reason does not appear in the sidebar list of “Recent entries”.

Posted by: Simon Pepin Lehalleur on October 27, 2022 9:24 AM | Permalink | Reply to this

### Re: Lack of Title

I accidentally deleted the title while fixing the mistake Theo pointed out, but now it’s back. Thanks!

Posted by: John Baez on October 27, 2022 9:52 AM | Permalink | Reply to this

### Re: Booleans, Natural Numbers, Young Diagrams, Schur Functors

Stopped myself from exclaiming “Littlewood-Richardson” on remembering that Young Diagram Containment is not a total order. Oh well…

Posted by: Jesse C McKeown on October 31, 2022 12:08 PM | Permalink | Reply to this

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