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March 5, 2014

Operads of Finite Groups

Posted by Tom Leinster

Guest post by Nick Gurski

I have been thinking about various sorts of operads with my PhD student Alex Corner, and have become interested in the following very concrete question: what are examples of operads in the category of finite groups under the cartesian product? I don’t know any really interesting examples, but maybe you do! After the break I will explain why I got interested in this question, and tell you about some examples that I do know.

Alex and I started off thinking about various sorts of things you might do with operads in Cat\mathbf{Cat}, and were eventually forced into what we currently call an action operad. This is an operad GG whose job it is to act on the objects of other operads. The key examples to keep in mind are the terminal operad (each set is just a singleton), the symmetric operad (the nnth set is the nnth symmetric group), and the braid operad (the nnth set is the nnth braid group). The technical definition involves an operad GG, a group structure on each set G(n)G(n), a map of operads π:GΣ\pi:G \rightarrow \Sigma to the symmetric operad which is levelwise a group homomorphism, and a final condition (when it makes sense) relating operadic composition, μ\mu, with group multiplication:

μ(g;f 1,,f n)μ(g;f 1,,f n)=μ(gg;f π(g)(1)f 1,,f π(g)(n)f n). \mu(g; f_1, \ldots, f_n) \cdot \mu(g'; f_1', \ldots, f_n') = \mu (g g'; f_{\pi(g')(1)}f_{1}', \ldots, f_{\pi(g')(n)}f_{n}').

These ideas have cropped up before, for example in Nathalie Wahl’s thesis or this preprint of Wenbin Zhang. Once you have this definition, you can define operads which are equivariant with respect to GG: you have an operad PP, a group action of G(n)G(n) on P(n)P(n) for each nn, and some equivariance conditions that generalize the equivariance conditions for a symmetric operad.

This isn’t the only thing you can do with an action operad, you can also think about the 2-monad on Cat\mathbf{Cat} whose algebras are strict monoidal categories where G(n)G(n) acts naturally on nn-fold tensor products. If you do this with the symmetric operad, you get symmetric strict monoidal categories (or permutative categories, if you are a topologist); if you do this with the (ribbon) braid operad, you get (ribbon) braided strict monoidal categories; and if you do this with the action operad of all terminal groups, you get back plain old strict monoidal categories. The only other “naturally-occurring” example of an action operad that I know of is the operad of nn-fruit cactus groups, J nJ_{n}. These groups come up in the representation theory of quantum groups, particularly the theory of crystals, and the monoidal structure you get out here is something Drinfeld called a coboundary category. I can give you a generators-and-relations definition of these groups, at which point I would have completely exhausted my understanding of this operad. The best reference that I know of is the paper Crystals and coboundary categories by Henriques and Kamnitzer.

What does this have to do with my question about operads of finite groups? Well, as it turns out, the structure map for an action operad π:GΣ\pi:G \rightarrow \Sigma only has two options: it can be surjective, or it can be the zero map (i.e., everything maps to the identity permutation). Furthermore, that condition I wrote down relating group multiplication and operadic composition says that giving an action operad with π\pi the zero map is equivalent to giving an operad in which the operadic composition maps all preserve group multiplication. Alex and I already showed that the operadic composition of all identity elements is the identity element in the target, in other words an action operad with π\pi the zero map is just an operad in the category of groups using the cartesian product.

You can take kernels of maps between action operads, so in particular given any action operad GG you can take the kernel of π:GΣ\pi:G \rightarrow \Sigma; this gives you an operad in groups. For the examples above, you get finite groups when GG is the terminal operad or G=ΣG = \Sigma (as the terminal operad is obviously the kernel in the case of G=ΣG = \Sigma), but for the rest of the examples you get an operad in the category of groups, but most of those groups are infinite. The groups involved are the so-called pure versions: pure braids, pure ribbon braids, and pure nn-fruit cacti. One can then think of an action operad with surjective map π\pi as being an extension of the symmetric operad by an operad in the category of groups (the “pure” version), and that action operad is finite if and only if the operad in the category of groups is one containing only finite groups. We can translate this back into thinking about monoidal categories by then noting if we have some notion of strict monoidal category in which nnth tensor powers come equipped with a natural action of a finite group G(n)G(n) for all nn, then we must be able to dig up an operad in the category of finite groups.

Now let’s talk concrete examples: what operads do I know in the category of finite groups? Well, there is obviously the terminal operad, but I can go very slightly further in that I can tell you how to construct some new ones. Here are two methods you can use to construct operads of finite groups.

  • Let AA be a finite abelian group. Then there is an operad A̲\underline{A} where A̲(n)=A n\underline{A}(n) = A^{n} (the power here is a cartesian power). The operadic composition map A n×A k 1××A k nA Σk i A^{n} \times A^{k_{1}} \times \cdots \times A^{k_{n}} \rightarrow A^{\Sigma k_{i}} takes the vector (a 1,,a n)(a_{1}, \ldots, a_{n}) in the first coordinate and duplicates the iith coordinate k ik_{i} times, then adds the result to the vector you get by just concatenating the nn vectors in the A k iA^{k_{i}}. Here AA must be abelian as it appears as A̲(1)\underline{A}(1), and G(1)G(1) must be abelian for any action operad by an Eckmann-Hilton argument.
  • Now let GG be any finite group, but in fact you will see that we don’t use anything interesting about the group structure here. There is an operad G c2G^{c2} with G c2(n)=G (n2) G^{c2}(n) = G^{\binom{n}{2}} given the pointwise group structure. You should think of this as the set of functions from {(i,j):1i<jn}\{ (i,j): 1 \leq i \lt j \leq n \} to GG. Operad composition is a map G (n2)×G (k 12)××G (k n2)G (Σk i2). G^{\binom{n}{2}} \times G^{\binom{k_{1}}{2}} \times \cdots \times G^{\binom{k_{n}}{2}} \rightarrow G^{\binom{\Sigma k_{i}}{2}}. If we are given 1a<bk i1 \leq a \lt b \leq \sum k_{i}, we must give back an element of GG. If there is some rr such that i=1 r1k ia<b< i=1 rk i, \sum_{i=1}^{r-1} k_{i} \leq a \lt b \lt \sum_{i=1}^{r} k_{i}, then we use the function coming from G c2(r)G^{c2}(r) evaluated on 1a+1 i=1 r1k i<b+1 i=1 r1k ik r. 1 \leq a +1 - \sum_{i=1}^{r-1} k_{i} \lt b+1- \sum_{i=1}^{r-1} k_{i} \leq k_{r}. If not, then there exist r<sr \lt s such that aa lies in the rrth “interval” as we had before and bb lies in the ssth interval, so then you use the function coming from G c2(n)G^{c2}(n) evaluated on r<sr \lt s.

I am reasonably content with the first of these constructions, I understand how to do the second but don’t really know where it comes from, and I have some ideas about how one could try to insert the finite group of their choice as G(0)G(0) but haven’t checked the details, and know basically nothing else. Furthermore, I don’t know how these interact with each other, or how you can form extensions of Σ\Sigma with them outside of some obvious constructions.

Those are just the two straightforward approaches that I know of to construct operads in the category of finite groups. You can also try to construct these operads from operads of topological spaces or simplicial sets, but once again I don’t know of an example that produces finite things (apart from ones giving the groups above). Do you know any others?

Posted at March 5, 2014 4:48 PM UTC

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Re: Operads of Finite Groups

The second construction seems to come from dualizing a cooperad in (Set,)(\text{Set}, \sqcup). I don’t have anything intelligent to say about cooperads, but if you found more of these you would get more examples for finite groups, but they would also not really use the group structure.

Posted by: Qiaochu Yuan on March 5, 2014 7:16 PM | Permalink | Reply to this

Re: Operads of Finite Groups

I actually tried for a little while to think about both of these constructions as homming out of a cooperad in the category of sets, but gave up because the first one clearly isn’t. The counit forces any cooperad to have C(1)C(1) empty, and if the construction AA̲A \mapsto \underline{A} is dual to some cooperad then you would need C(1)C(1) to be a singleton.

Now the second construction does work this way, as you point out, and the very rough idea I sketched basically gives C(n)=(n2) C(n) = \binom{n}{2} a cooperad structure. One of my secret hopes for this post was that some combinatorially-savvy reader would come along and explain cooperads in finite sets.

Posted by: Nick Gurski on March 6, 2014 11:15 AM | Permalink | Reply to this

Re: Operads of Finite Groups

Ah, excellent, I spent a lot of time thinking about operads in groups as a PhD student before settling on something different to write for my thesis.

Ofcourse the first question should be why groups, and not groupoids, or monoids, or categories?

I think the groupoid perspective is most important, partly because the free operad is so much nicer. One could then fix a set operad of one’s choice and then ask for groupoid operads over that set operad. For example, Vladimir Dotsenko and I have found a family of examples (non-finite) over the permutative operad: http://arxiv.org/abs/1109.5345 (though I stress that we didn’t fix the operad of points first!) Fixing the commutative operad one arrives at symmetric operads in groups. The associative operad is another interesting case.

Another reason for considering groupoids is one of the most important examples, the pure braid groups. Yes, these might form a non-symmetric operad in groups but that is ignoring important structure. They can’t be made into a symmetric operad in groups, but they are a symmetric operad in groupoids.

Anyway, that’s not what you asked for, you wanted a source of operads in finite groups. For this you can just use your discussion about actions on monoidal categories, but reverse it. Fix a finite Hopf algebra HH, e.g. a finite group algebra over a finite field. Now consider natural transformations from the n-fold tensor product functor to itself, call them 𝒪 H(n)\mathcal{O}_H(n). This is an operad in the category of monoids, take invertible elements if that’s your wish.

Through a quick computation you find that 𝒪 H(n)=C H n(Δ(H)),\mathcal{O}_H(n) = C_{H^{\otimes n}}(\Delta(H)), where Δ(H)\Delta(H) is the image of n-fold coproduct and “C” stands for centralizer. I was able to compute some of these monoids/groups in low arities by hand, though did not get very far in answering even simple questions, such as if these operads could be finitely generated. I’m sure that proper use of a computer would help to get a better handle on these.

Moving out of the finite world there are other special classes of operads in connected groupoids (not another term for groups). I have never understood why no one has looked into this more, it’s perfect PhD fodder.

Posted by: James Griffin on March 5, 2014 8:25 PM | Permalink | Reply to this

Re: Operads of Finite Groups

Alex and I thought a little bit about using monoids instead of groups, but didn’t because of the standard reason: groups are easier. Almost every formula that we ran into had either a gg on one side or a g 1g^{-1} on the other, and using groups means you can be a little sloppy about which you actually take as fundamental. That seems like a compelling reason when you are trying to wrap something up, and is a terrible reason mathematically. Someone could surely go through a lot of these operadic constructions and figure out when you can use monoids of equivariance instead of groups, and honestly probably learn a lot in the process.

Thanks for the family of examples, that is exactly the sort of thing I hoped someone would come up with!

Posted by: Nick Gurski on March 6, 2014 11:40 AM | Permalink | Reply to this

Re: Operads of Finite Groups

No problem. And I absolutely take your point about working with groups, especially if you ever have to do anything with them.

Posted by: James Griffin on March 8, 2014 3:48 PM | Permalink | Reply to this

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