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Unfortunately, I need to discuss with you another terminological problem. I am lightly doing a circle of entries related to combinatorial aspects of representation theory. I stumbled accross permutation representation entry. It says that the permutation representation is the representation in category $Set$. Well, nice but not that standard among representation theorists themselves. Over there one takes such a thing – representation by permutations of a finite group $G$ on a set $X$, and looks what happens in the vector space of functions into a field $K$. As we know, for a group element $g$ the definition is, $(g f)(x) = f(g^{-1} x)$, for $f: X\to K$ is the way to induce a representation on the function space $K^X$. The latter representation is called the permutation representation in the standard representation theory books like in
I know what to do approximately, we should probably keep both notions in the entry (and be careful when refering to this page – do we mean representation by permutations, what is current content or permutation representation in the rep. theory on vector spaces sense). But maybe people (Todd?) have some experience with this terminology.
Edit: new (related) entries for Claudio Procesi and Arun Ram.
FWIW, I would have assumed “permutation representation” to mean in the standard representation-theory sense of vector spaces.
I’d be inclined to call the current topic of the page a representation by permutations. I note here that $C$ should be a groupoid; otherwise “permutation” is completely out.
Right I do call it “representation by permutations” (see 1), while the permutation representation in the rep. theory sense entails the linear extension of the action from a vector space basis to the entire linear (=vector) space.
Zoran #5: I am in agreement with this terminology. When I talk privately with certain category theorists (e.g., Jim Dolan), sometimes we’ll refer to a permutation representation when we mean a representation in $Set$ (which latter by the way is also acceptable), but this is not widely recognized outside a certain community.
Jim #3: true, but the rest of the world has apparently not caught up yet! When people say ’representation’ without qualifications, the default assumption is that it’s a linear representation, and a permutation representation is generally taken to mean a composite functor
$B G \to Set \stackrel{free}{\to} Vect_k.$It makes sense to say “permutation representation” for $\cdots \to \mathrm{Set}$ and “linear permutation representation” for $\cdots \to \mathrm{Set} \to \mathrm{Vect}$.
Sure, it makes sense to say that, but if you only said “permutation representation” by itself to the mathematician-in-the-street, he is likely to interpret that as what you just called “linear permutation representation”. I thought that’s what this discussion was about: what the conventional standard is (as opposed to what the convention should be), as reported in Zoran’s #1.
I thought that’s what this discussion was about: what the conventional standard is
Okay. I thought the discussion is about what to write in the $n$Lab entry.
Of course we should always explain different use of terminology, for transparency. But I don’t think that we have to enslave ourselves to conventional standards if they should be different, as in this case.
I have added a word to the entry.
One difference between a representation in $Set$ and a permutation representation in $Vect$ is the morphisms (the intertwiners). Even the isomorphisms differ, so we can’t pretend that they’re the same thing.
they should be different, as in this case
Clear “representation by permutations” for categorical notion and cryptic “permutation representation” for linear notion are different.
I should point out that people who work on representations of symmetric groups and applications in combinatorics refer to permutation representations although there is often not a linear representation in sight. This continues the tradition of D.E. Littlewood. (You will see how I know this if you look at http://en.wikipedia.org/wiki/Dudley_E._Littlewood).
added a remark Comparison map from Burnside ring to representation ring
starting a section Examples – Virtual permutation representations. But we happen to be discussing this not here in this thread, but in the thread on the Burnside ring, see there
In the proof of the proposition about $\beta$ being surjective for some classes of groups (this prop.), I have expanded out the argument for cyclic groups, as a corollary of that for $p$-primary groups. Like so:
To see surjectivity for cyclic groups: By the previous statement we have surjectivity already for those cyclic groups whose order is a prime power. But by the fundamental theorem of cyclic groups, every cyclic group is a direct product group of cyclic groups of prime power order. Moreover, every irreducible representation of a direct product group is an external tensor product of irreps of the group factors (this prop.). But $\beta$ sends “external Cartesian products” to external tensor products, by the same elementary argument which shows that $\beta$ sends plain Cartesian products to tensor products. This way the statement reduces to that for $p$-primary cyclic groups.
Hm, maybe my argument for the cyclic groups (#18) is wrong after all:
That the irrep of a direct product group is a tensor product of irreps, is that true also over $\mathbb{Q}$?
[ edit: hm, should be okay, as in [prop. 2.3.17 here]
Ah, the surjectivity of beta for cyclic groups over the rationals is example 4.4.4 in tom Dieck’s notes http://www.uni-math.gwdg.de/tammo/rep.pdf
added pointer to Dress 86, section 3 for the statement that $\beta$ is surjective for symmetric groups in characteristic zero.
Re #24: well, that is indeed wonderful since I didn’t know where to find it in the literature. Now I do.
It’s a citation, but still not a proof, since it just says
it follows easily from the classical representation theory of $\Sigma_n$…
What is indeed easy to see is that in the discussion of Specht modules all the ingredients enter that one expects to see for this to be true. But if it’s so easy, what is the explicit general formula that expresses the Specht module corresponding to a given Young tabloid in terms of virtual permutation reps?
I might ask on the group-pub mailing list (which, for some weird reason, is private) to see if this is treated explicitly anywhere.
Didn’t know of this group. If you can, please do!
So David kindly checked on that group theory list. In reply there is so far a sketch of the idea of the proof:
if you look in the representation theory of $S_n$, the situation is the following:
you have an ordering on the set of partitions of $n$, which is, if I am not mistaken, the lexicographical ordering.
Then for every partition $\lambda$, you define $M_\lambda = Ind_{S_{\lambda}}^S_n 1$.
Inside $M_{\lambda}$ you find a direct sum of a new irreducible module $V_{\lambda}$ and modules of the form $V_{\mu}$, for $\mu \lt \lambda$.
This already shows that if you consider the representing matrix of the above transformation, you get an upper diagonal matrix, with 1s on the diagonal.
(In $A(S_n)$ I have restricted here to subgroups of the form $S_{\lambda} = \prod_{i=1}^r S_{\lambda_i}$)
and, upon further request for a citable reference, a pointer to Sagan 01 (but it seems that’s meant as a general pointer to the rep theory of the symmetric group, not to a citable proof of the statement in question).
Okay, thanks Urs and David – I’ll check through that later. One of those things that is absolutely routine to experts, it seems.
Well, thinking about what was written for about 90 seconds, this
Inside $M_{\lambda}$ you find a direct sum of a new irreducible module $V_{\lambda}$ and modules of the form $V_{\mu}$, for $\mu \lt \lambda$
looks pretty hand-wavy: about as hand-wavy as the Gram-Schmidt page. But maybe it will become clearer in time. If I get desperate, maybe I’ll put it to MO.
Thanks, Todd, for looking into this.
(The situation reminds me of what Langlands said (somewhere here) about specialists hurting their field by being so specialist.)
Here is a first list of our results (using computer algebra by Simon Burton) on cokernels of $A(G) \overset{\beta}{\to} R^{(fin)}_k(G)$ for nonabelian finite subgroups of $SU(2)$: jpg.
In all cases, beta is surjective over $\mathbb{R}$ onto the ring of integer-valued (i.e. not irrational-valued) characters.
Interesting how many items in that right-most column are essentially $\mathbb{Z}[\rho]/\mathbb{Z}[2\rho]$. Is that due to the central extensions by $\mathbb{Z}/2$?
Don’t know if this is related to any central extension. Here one such contribution appears for every complex irrep with integer characters but of quaternionic/symplectic type.
I’ll show the details of the computation later when the file has taken shape a bit more.
For what it’s worth, here is an overview of the physics picture behind that result in #33: jpg
If we see group characters appear in for fractional D-branes, should we not expect marks to play a role in M-theory?
Much like character theory simplifies working with group representations, marks simplify working with permutation representations and the Burnside ring. (Wikipedia)
According to Pfeiffer here:
The table of marks of $G$ arises from a characterization of the permutation representations of $G$ by certain numbers of fixed points. It provides a compact description of the subgroup lattice of $G$ and enables explicit calculations in the Burnside ring of $G$.
Plausibly, but I remain uncertain what exactly there is to be said.
A special kind of mark, namely that of cyclic groups $C_g$ generated by elements $g \in G$, already does play a central role, because that is equivalently just the character of a permutation representation $k[G/H]$ (for any ground field $k$):
$m(H, C_g ) \;=\; \chi_{k[G/H]}(g) \,.$Here is finally the writeup of the result mentioned in #33 (no changes since then, really, but now we decided to just wrap it up):
S. Burton, H. Sati, U. Schreiber,
The image of the Burnside ring in the Representation ring – For binary Platonic groups
Abstract We describe an efficient algorithm that computes, for any finite group $G$, the linear span of its virtual permutation representations inside all its linear representations, hence the image of the canonical morphism $A(G) \overset{\beta}{\longrightarrow} R_k(G)$ from the Burnside ring to the representation ring. We use this to determine the image and cokernel of $\beta$ for binary Platonic groups, hence for finite subgroups of $SU(2)$, over $k \in \{\mathbb{Q}, \mathbb{R}, \mathbb{C}\}$. We find explicitly that for the three exceptional subgroups (2T,2I, 2O) and for the first seven binary dihedral groups, $\beta$ surjects onto the sub-lattice $R^{\mathrm{int}}_{\mathbb{R}}(G)$ of the real representation ring spanned by the integer-valued characters. We conjecture that, generally, this holds true for all the binary dihedral groups.
Comments are welcome.
What was the scope of the research that Todd and James Dolan were talking about? Over the years, there has been talk of many large projects involving James and others, with mention of buildings, groupoidification, doctrines. Is it possible to give a sense of the ambitions of this work?
Just in case the question is aimed at me, I’ll say that I have no idea what James Dolan has been talking about with anyone, as I was never involved.
The acknowledgement in our writeup from comment #39 above – if that’s what prompted your question in #40 – is just saying that the algorithm we discuss was inspired by Todd’s note from 2009 here, which in its first sentence acknowledges, in turn, a suggestion by James Dolan.
Just saying, in case the question was aimed at me, written seemingly in reply to my announcement in #39.
David, can you give the referent for “research that Todd and James Dolan were talking about”?
I never began to get the scope of whatever it was Jim et al. were driving at in that period. I take it that whatever it was, it wasn’t exhausted by the Groupoidification series, or the seminars, Fall 2007 and Winter 2008, and these ranged far and wide over Geometric Representation Theory. I think the Doctrines project was part of it.
It was clearly meant to be much more that the provider of some calculational tool for virtual permutation representations. Urs wrote
For the longest time the “groupoidification” program looked like idle entertainment to me, but now that I understand the importance for fundamental physics of computing the image of equivariant stable cohomotopy in equivariant complex K-theory, I have a different feeling for it.
There were surely greater expectations. Since nobody works on it now, it would good to have some record of what the ambitions for this project were. Somebody might want to pick up the pieces one day.
I thought, Todd, you had a good idea where Jim was heading, as indicated by your writing
a whole slew of interesting developments, in which we view Jim’s orbi-simplex idea as a geometric description of a general axiomatic theory, which in turn is related to the idea of viewing Tits buildings as “quantized” axiomatic theories, and also perhaps to the theory of classifying toposes and their “Galois theory”. But we’ll get to all that later!
Thanks, David. Let me get back to you on this. I just didn’t know where you were looking (as in, where were Jim and I talking about this).
I just wrote Jim an email today. I’m hoping he can help resurrect some memories, because a lot of what we were discussing more than 10 years ago was never properly written down and begins to fade from my memory, but it shouldn’t be impossible for us to piece it together, and then I can more easily report back here. There were some very interesting “groupoidified” quantum group calculations which I’d really like to retrieve, for example.
The categorical treatment of Tits buildings I can recall large parts of, but not necessarily the meta-thoughts Jim had in mind more programmatically.
$[$ Groupoidification$]$ was clearly meant to be much more that the provider of some calculational tool for virtual permutation representations.
I didn’t see anything about “groupoidification” being a calculational tool for virtual permutation representations in the first place. Do you have a pointer?
Re #46, I didn’t mean that it was explicitly stated somewhere that being a calculational tool was an intention, though they certainly wanted to see how far they could get using combinatorics rather than linear algebra over vector spaces.
what’s groupoidification? It’s a method of exposing the combinatorial underpinnings of linear algebra - the hard bones of set theory underlying the flexibility of the continuum.
The issue I’m raising is how to look back on the program. If it’s only valuable influence is Jim conveying to Todd some ideas on categorified Gram-Schmidt, which later are taken up by you looking at what persists in K-theory from some M-theoretic construction, that’s better than nothing. But it was presented once as something very grand:
If we let this story lead us where it wants to go, we’ll meet all sorts of famous and fascinating creatures, such as:
- Coxeter groups, buildings, and the quantization of logic
- Hecke algebras and Hecke operators
- categorified quantum groups and Khovanov homology
- Kleinian singularities and the McKay correspondence
- quiver representations and Hall algebras
- intersection cohomology, perverse sheaves and Kazhdan-Lusztig theory
However, the charm of the tale is how many of these ideas are unified and made simpler thanks to a big, simple idea: groupoidification.
And I believe Jim had bigger plans, such as his work on Doctrines.
I didn’t mean that it was explicitly stated somewhere that being a calculational tool was an intention, though they certainly wanted to see how far they could get using combinatorics rather than linear algebra over vector spaces.
Sure, but did “groupoidification” ever say anything about virtual permutation representations? I checked the article again, and I see not a hint of that.
Incidentally, what did “groupoidification” achieve? What’s a list of results? I mean actual results, beyond observing that pull-push through spans of tame groupoids makes sense. I remember at some point there was the claim of a theorem that after splitting idempotents the image of the linearization map from tame spans to vector spaces is all of finite dimensional vector spaces. Has that been proven? I don’t see that mentioned in the available articles.
I don’t see that this conversation is going to achieve much the way it’s currently being conducted, without some input either from me or John or Jim or Alex Hoffnung or maybe someone else. A lot of the tone is pretty off-putting to me. Is it really necessary to use the sneer quotes in “groupoidification” each and every time?
Speaking to Urs in #48, I can for example point to this concrete result we obtained: that a $W$-building $(X, d)$ in the sense of Tits can be identified with a strong $\ast$-monoidal closed functor of the form
$2^d: 2^W \to 2^{X \times X}$between $\ast$-quantales, where $2^W$ carries Day convolution from the $\ast$-monoid structure on the Bruhat-Tits monoid $W$, and $2^{X \times X}$ is the usual $\ast$-monoid of binary relations on the set $X$.
If you give me a bit of time, I can recall some more calculations which I think are interesting.
Sorry, I didn’t mean to stir things up. But now it’s been pointed out, I don’t recall that virtual permutation reps were considered, and they are needed for the combinatorial to get anywhere near the linear algebra.
True. And I do mean to look further at this topic (and was, except that I was interrupted recently by having to prepare for a job interview, which has now been completed). John and Simon and I were talking a bit about this behind the scenes, with an eye especially toward a proper understanding of Young symmetrizers.
Todd,
interesting how the perception differs. I found it remarkable how my #39 triggered a whole shower of praise for “groupoidification”, suggesting that, while unfortunately not having materialized, it is “much more” than what I had mentioned. If that is the case, I’d like to and need to know what that is, in particular to cite properly. This requires me trying to break through the veil of vague indications of greatness and ask concrete questions, such as “What did ’groupoidification’ achieve?” This is the most reasonable question to ask of any research program, and perfectly matter-of-fact. I am or will be grateful for replies (and also our entry groupoidification would deserve to be given some concrete information of this kind).
Is it really necessary to use the sneer quotes in “groupoidification” each and every time?
I find it really important in a public context where a) the term categorification has a fixed default meaning and b) newbie lurkers are expected to be following discussions – so as to carry a minimum of warning that what is called here “groupoidification” is not the general concept of categorification with categories taken to be groupoids, but is something rather different and much more specific (a kind of lift through a linearization map). I always felt it adds to the mystery and mystique surrounding the program that its name is not only unsuggestive of its content, but in tension with the tacitly understood perspective held by its practitioners.
To wit, while I am thankful for you replying to “What did ’groupoidification’ achieve?”, I am currently at a loss to see how the statement you mention in #49 is an example or application of “groupoidification” as laid out in Baez-Hoffnung-Walker ’s arXiv:0908.4305.
Maybe, instead, the first question I should ask you is: What is “groupoidification”? Or if you get sick of my quotation marks and if it isn’t really so much about groupoids or groupoidification anyway, let’s maybe call it “Dolan-infication” and ask: What is Dolan-infication? In your words.
Okay, I don’t even know what’s in Baez-Hoffnung-Walker.
I don’t want to call whatever it is Dolanification.
But, I will be talking to Jim about this to get his take. I think he’s probably the best person to ask about the hopes and dreams, and maybe some concrete achievements too, from that particular era.
it is “much more” than what I had mentioned
Sorry, what are you referring to here? Is the mention a particular comment you made, or something in a paper, or…?
But if you can tell me specifically what you need for the purposes of citation, I can try to do my best to help.
For some reason I was not picking up on showers of praise, but I was picking up on skepticism that the project ever amounted to anything beyond grandiose statements. (Which would be understandable from the perspective of publications that emerged, I guess.) But it might be interesting trying to articulate what the scope was supposed to be – perhaps in a separate thread.
Please give me a few days, seeing that it’s Christmas and so on.
Sorry, what are you referring to here? Is the mention a particular comment you made, or something in a paper, or…?
But if you can tell me specifically what you need for the purposes of citation, I can try to do my best to help.
The discussion we are having was triggered by my #39. In #43 David suggested that the Trimble-Dolan program is “clearly much more”. I was surprised to hear that in #46, and clarified my point in #48.
As far as I can tell we are citing quite generously in the present version. I’ll be happy to cite more where necessary. But we should avoid a situation where new insights can’t be communicated over a diffuse noise of buzzword dropping.
The abstract you quoted in #39 doesn’t mention groupoidification. Is it something you wanted to discuss in the paper?
If not, then it sounds like there’s no hurry to discuss what the term is understood to comprise by one or another of its proponents. I didn’t think it had a precisely delineated meaning. (Getting Jim to precisely delineate a meaning is, in case you didn’t know already, often extremely difficult.) I could try to talk about it, but: at greater leisure, and in a separate thread.
If you do want to discuss groupoidification in the paper, just let me know if I can be of any help. But it doesn’t sound like that’s really the case.
As I understand it, David is right that the term connotes, at least in Jim’s mind, a vast territory including the topics that he (David) quoted from John. But that doesn’t mean that a great number of details had been worked out. For the sake of the newbies and lurkers out there, it shouldn’t be referred to here as the “Trimble-Dolan program”, as that could set up expectations that I can’t promise delivery for. However, all this stuff is of latent interest to me personally, and something I’d like to get back to.
Let’s be more concrete.
I still seem to remember there was a claim, or maybe a speculation, that the functor from finite groupoids with tame spans between them to finite-dimensional vector spaces over a suitable field exhibits idempotent completion, or something close.
Has something like this been proven?
Don’t know; if I ever knew what it means for a span to be tame, I’ve forgotten (could you tell me?). But it sounds like could be interesting to consider.
@Todd
I believe tame essentially meant standard pushforward (sum over fibres) was well-defined. So something like appropriately finite fibres. See Definition 13 of HDA VII.
David, in #53 Todd said that he “doesn’t know” what is in HDA VII.
Since this text HDA VII is course the 9 year old standard (and essentially only) reference on the topic of “groupoidification” with that functor from tame spans being the center of attention (it carries essentially the name of the program), and since Todd must very well be aware of its existence since at least 9 years ago (preceded and accompanied, as it was, by an extensive discussion in lectures and blog entries, as Google helps remembering), the conclusion I draw now is that when Todd says “groupoidification” he wants it to mean something really different from what John Baez advertized it to be.
Is that right, Todd?
But then, what is it? What is it we are discussing? On my end now, what I am left with is a list of buzzwords as quoted in #47 with no positive indication as to what claim there is.
I had voiced this already in #52, asking Todd to say what he wants “groupoidification” to mean. I take the reply in #53 to say that Todd can’t answer this, since it’s only Jim Dolan who would know. Only that Jim will never quite say it either, right?
@Urs,
I was responding to the direct request:
if I ever knew what it means for a span to be tame, I’ve forgotten (could you tell me?).
with a pointer to the definition, not anything bigger than that. I was being a little lazy and not writing it out here, because there was a (short) chain of definitions that needed tracing back in order to make sense of the definition of a “tame span of groupoids”, and I didn’t have time to track them down and write them up (big and exciting change in my family :-), so less time than usual around this time of the year for mathematics).
No, I wasn’t really aware of its existence. Or at best only dimly aware. That’s the truth. I hope I’m not going to come under attack for circumstances from a long time ago.
I can try to put together a response to “what is groupoidification?”, but it would take some time. Urs, if you would like to continue the discussion, it would be better to be patient and not to jump to conclusions publicly. I’m getting an awful lot of hostility or aggression from you at this point, and it’s going to be hard for me to continue under those circumstances.
Hi Todd,
it’s a shame that you perceive plain requests for mathematical content as aggression.
I’d really like to know what’s going on. It is the strangest situation we have here.
But of course I agree that the discussion is not leading anywhere. I don’t think this is can sensibly be blamed on me, but, fine, let’s leave it at that.
In the Prop. (here) on examples where $A(G) \overset{\beta}{\to} R_{\mathbb{Q}}(G)$ is surjective, I have added the cases of binary dihedral groups, of $2I$, $2O$, $2T$ and of $GL(2,\mathbb{F}_3)$, referencing the brute force computation in our preprint Burton-Sati-Schreiber 18
I was hoping my email to the group-pub mailing list pointing out the conjecture in your paper (about the binary dihedral groups in general) would get some response, but, sadly, nothing.
Thanks David. That group hasn’t been all that responsive, throughout, but thanks for trying. I should go and subscribe there myself.
I see some chance that I can convince Simon to add just a little to the Python script such as to make it automatically call the character tables from GAP and match them to the basis spanning the image of beta. This would make the computation entirely automatic (while currently the reason we don’t include many more example is that for each one I have to tediously look up characters and read off coefficients by hand, which, quite seriously, has been hurting my eyes.)
With a full automatization we could check the first, say, ten thousand binary dihedral groups, which should make the result more interesting in itself (for those readers who don’t see yet the striking implications for fractional D-brane physics…).
If you have the existing source available and Simon does not time, I can probably do it for you.
Hi Richard,
that would be awesome if you would join in. I am convinced that there is potential here for very interesting “experimental mathematics”:
Given that we see in more and more examples that $\beta$ is surjective, or at least integrally surjective – which is a most interesting situation – it is suggestive that this will hold in very large classes of further examples. In the absence of any theory of why this is so, it should be most interesting to do computer scans of large classes of further examples.
Simon’s Python code as used in our article is available at arxiv.org/src/1812.09679v1/anc, which is, I think, a snapshot of his GitHub repository here. If or when you are serious about looking into it, best if we email Simon to inform him and bring you two in contact, to avoid clashes.
By the way, there are further immediate things that would be most interesting to implement. Next interesting would be to have a computer compute the Chern characters of the representations in/not in the image of $\beta$, in the sense discussed here. Conjecturally the image of $\beta$ would pick out those reps whose Chern characters is bounded in degree to $\leq 10$ and be Hodge self-dual in an appropriate sense.
Just saying, in case you are interested in more of a project. No lack of ideas for what to explore here and good chances for interesting results.
Hi Urs, great! In principle I am very interested! I don’t know much about the mathematics, but I am happy to try to help with the experiments.
There are several things that I would like to do with the nLab code at the moment. But I may be able to do the thing you need with GAP quickly. I could probably also make a little user interface here on the nLab server if that would be convenient, and if one can imagine others being interested in these computations.
Could you maybe explain exactly what you do by hand (including code samples if possible) at the moment? I should be able to then quickly collect it into a script for you. For the moment I can just make the script outside of Simon’s ones, just calling them; then Simon can choose if he wishes to incorporate them.
Thanks, okay. Here is how we used to proceed:
Let’s look at the pdf here, the first example, section 4.1, pages 16-17. Simon’s code computes the stuff on p. 16 and ends with computing the first table on the top of p. 17, which is the table of characters of a list of representations $V_i$ (which we know to span the image of $\beta$).
The rest of that page 17 is what I do by hand: First I look up the tables of characters of the irreducible (complex and real) representations $\rho_i$. By comparing these with the table of characters of the $V_i$, I deduce the expansion of the $V_i$ as linear combinations of the $\rho_i$.
This would be the first thing that should be further automated: Given the list of the vectors (characters) $V_i$:
make GAP spit out the list of basis vectors (irreducible characters) $\rho_i$
express the $V_i$ as linear combinations of the $\rho_i$.
Finally, on the bottom of that page 17 I sum this up by giving the cokernel of $\beta$, namely the abelian group spanned by the $\rho_i$ quotiented by the abelian subgroup spanned by the $V_i$.
This is not strictly neccessary to be automated in this form. What we really want to know in the end is two yes/no questions:
Do the $V_i$ generate the same abelian group as the $\rho_i$? (i.e. is $\beta$-surjective?)
If not, do they at least generate the same abelian group as those $\rho_i$ whose characters have components that are integers (equivalently: are not irrational)?
So this means:
Is the integer matrix that expresses the $V_i$ as a linear combination of the $\rho_i$ invertible?
If not, is it invertible after discarding those columns that correspond to non-integer (equivalently irrational) characters?
This should all be fairly trivial. I imagine the only part that may need care is to figure out how to match the conjugacy classes of subgroups (which label the entries of characters) between Simon’s code and GAP.
I know GAP only from hearsay, so I don’t know how this will work.
There may be a major shortcut:
The matrix $mult_{i j}$ or $M_{i j}$ of “total Burnside multiplicities” is equivalently the table of marks of the group, and that may be just called directly from GAP (here).
(It is my silly mistake for not knowing that GAP can do this right away. I didn’t look into the coding part of the problem for too long.)
This should allow to rewrite the full algorithm in streamlined form from scratch:
Call the table of marks $M$ from GAP.
Apply “simple-minded” row reduction $\tilde H = \tilde U \cdot M$ to it (as per Remark 3.26)
Compute the characters of the resulting rows $V_i$ (by $\chi_{V_i}([g]) = \tilde U_{i}^\ell \left \vert (G/H_i)^{\langle g\rangle} \right \vert$)
Call the character table of complex or real irreps $\rho_i$ from GAP.
Find the linear transformation expressing the (characters of the) $V_i$ in terms of the (characters of the) $\rho_i$.
Thanks very much, Urs, that looks like what I was looking for. I have used GAP before, but not for a number of years; still, it shouldn’t be a problem.
Unfortunately I do not have time today, but will try to have a look some time over the coming days if I get the chance. Just let me know if Simon does it before then.
So Urs (#62), I don’t think it would be worth my while to explain how I found you were being rude in #59, among other places. I’m happy just to let that ship sail by and move on.
I’m not planning on applying for a grant to support the groupoidification project any time soon, but maybe I could try to give a sense of the project – but at some point, and in a thread different from this. Naturally it would not consist only of “buzzwords” (a word you use pejoratively), but would give a proper sense in which the referents of those buzzwords find meaningful niches in the program – believe me, some of those topics came up in intensive conversations, years ago, and not just vaguely or in wishful thinking, but in detailed calculations. As I said before, it would take time to put such a description together. It’s not that I “can’t” do this – except I might need some help putting it together, partly because it was a while ago.
However, putting that on hold for the time being, it seems that there are specific problems you have in mind where some of us might put our heads together. Perhaps you’d be more impressed if some of those problems were solved. You seemed curious for example where you wrote
I remember at some point there was the claim of a theorem that after splitting idempotents the image of the linearization map from tame spans to vector spaces is all of finite dimensional vector spaces. Has that been proven? I don’t see that mentioned in the available articles.
So you thought you saw something to that effect somewhere? I actually read that aloud to Jim yesterday and he didn’t seem completely sure of what you were alluding to, but that something like that could easily have come up or might be plausible. (Taken literally, it doesn’t sound quite right to me, but I understand you’re just trying to convey an idea of the statement.) If something like this were of genuine interest to you, it’s something we could certainly think about.
It’s good that you are interested in using things like GAP to test conjectures. For myself, I’m not quite clear on the scope of categorified Gram-Schmidt, but I aim to find out. I assume this would be of theoretical interest to you as well. The material on the nLab page is utterly spindly and really needs to be amplified greatly – I’m planning to get to that.
More, I hope much more, later.
“buzzwords” (a word you use pejoratively)
Indeed. But as we found out, a core problem is that you think of something different when saying “groupoidification” than what we all think it refers to, based on the available articles and lecture notes with these titles. And maybe you can see why that is confusing (it caused us to talk completely past each other in most of the above discussion) and also irritating. Myself, I went through the trouble of checking the literature that goes with this keyword, then and again now, and I come back with the observation that it has grandiose announcements but is thin on substance. That’s why I told you, frankly, that without some minimum indication of what substance there is in “groupoidification”, it comes across as just buzzwords. And I still think it does. If this were a dating portal, saying this might count as rude, but since this is about mathematics I find it is necessary professional comment. If you do care about the term “groupoidification” being well received in public, as clearly you do, you might want to distinguish between the bearer of the bad news and the cause of the bad news. The first step might be to just use a different and more appropriate name for your program altogether.
I remember at some point there was the claim of a theorem that after splitting idempotents the image of the linearization map from tame spans to vector spaces is all of finite dimensional vector spaces. Has that been proven? I don’t see that mentioned in the available articles.
So you thought you saw something to that effect somewhere?
Yeah, I feel sure that John Baez did say something just like this back then. The other day I had spent a good while googling to see if I can recover the quote, but I get a flood of Google hits for the relevant keywords and couldn’t find the needle in the haystack.
But I don’t need this particular statement. I just think that for “groupoidification” as understood in the available literature to obtain substance, the first thing necessary is some characterization of that “degroupoidification” functor from tame spans to vector spaces. The available literature essentially ends with just pointing out that the degroupoidification functor exists. While that is a fun exercise, it really begs the question. Since this functor seems to want to be a finite toy analogue of the theory of Chow motives, the idea about idempotent completion seems natural. But if that’s not the appropriate perspective here, I’d like to know whatever else can be said about this functor.
I’m not quite clear on the scope of categorified Gram-Schmidt, but I aim to find out. I assume this would be of theoretical interest to you as well.
As you may have seen me say, I am interested in computing the cokernel and kernel of the canonical morphism of equivariant spectra $\mathbb{S}_G \to KO_G$, which on the point is the image of $A(G) \overset{\beta}{\to} R(G)$. For that the perspective of “categorified Gram-Schmidt” in itself does not seem to be that relevant, even though the actual computations are closely related. In fact, speaking of disentangling terminology, I had been thinking of asking you not to call “categorified Gram-Schmidt” what does not actually use Gram-Schmidt’s algorithm, but the Gaussian elimination algorithm. While the output of both algorithms turns out to be related and to coincide under suitable conditions, they are different processes. Also, I think the qualifier “categoriefied” seems inappropriate at this point, since you are looking at ordinary Gaussian elimination applied to a decategoried inner product. This is suggestive of raising the question if one can lift this through the decategorification map, but it’s not yet the answer to that question. I think accurate and suggestive terminology goes a long way. But okay.
Urs, as you know, things at the nLab can start off stubby and sketchy. Please regard “Categorified Gram-Schmidt” that way. Of course the intent all along is to eventually explain how to truly categorify this, but for now I am going to treat the title as a goad to doing this, and so I’d prefer holding off making changes to the title at the moment just to satisfy you.
I see what you mean about Gram-Schmidt vs. Gaussian, especially in that no actual “normalization” is performed in this particular application, but undoubtedly the current terminology is partly guided by the fact that we are starting off in this case with a basis for the representation ring and step by step we produce an orthonormal basis in the sense of 2-Hilbert spaces.
Re John: why not simply ask him? I can do this if you like. I do see the point you’re trying to make in this instance, and it’s a fair one.
bearer of the bad news and the cause of the bad news
Hm? Bad news? I don’t think there’s “bad news”, except for the fact that you seem to be strongly suggesting that the whole enterprise came to nought, which I don’t agree with. It’s just not been published to anywhere near the extent that it should have been by now. That’s the way it goes sometimes. Especially when some of the actors have not been employed and not that well connected.
Really, I don’t think you should be characterizing this as “bad news”. In fact, please don’t.
And once again, I really don’t feel I need to rush to explain to you, or anyone, what groupoidification or whatever one chooses to call it is about. It seems there is a disproportionate amount of hullabaloo in this thread over groupoidification that actually you don’t have an urgent need for, as far as I can tell. So in no way do I see this as a “necessary professional comment” – there is no need for you to harp on and on about this. Why not just wait and see? Time may tell.
All right. Just to say that we are talking about it because David brought it up seemingly in reaction to what i wrote and seemingly suggesting that I should be needing it. I tried not to ignore the comment and hence started asking questions.
I read David’s comment a bit differently – as usual the tone was one of gentle and open inquiry. But okay. It sounds like we’re ready to move on.
I read Urs’s comment back in September
For the longest time the “groupoidification” program looked like idle entertainment to me, but now that I understand the importance for fundamental physics of computing the image of equivariant stable cohomotopy in equivariant complex K-theory, I have a different feeling for it
as indicating a new interest in the program on his part, and with the reference in the new paper to Jim and Todd, and knowing that Todd still talks maths with Jim, I thought this might be a good moment to ask for an update from either. It seems not.
One would expect as a research project develops that it would be possible to sum up the approach, achievements and prospects. This case is unusual since the protagonists are outside (until now with Todd’s new job!) conventional mathematical careers. Were someone in post and encouraging the youth to look into their work, I should think there would be a duty to provide a honest appraisal of achievements and prospects.
To the extent that I now know that nobody here is just sitting on interesting insights into a larger picture without letting on, my question has been answered.
That’s right, my impression is that a way to make substantial sense of the idea of lifting linear algebra to combinatorics is to think of it as lifting through the map $\mathbb{S}_G \to K_G$. We have hard results saying that this indeed does live up to the promise, going back to none less than Segal 72; and we have good theory relating it to phenomena like the “quantization of logic” (that’s given by the formulation in terms of spectral Mackey functors) or “Kleinian singularities and the McKay correspondences” (that’s the modest contribution pointed to in #39) to take two items from your list of, let’s say keywords. In fact there is much, much more in equivariant stable (co)homotopy, with deep and wide relations to lots of maths, established or being worked out as we speak.
Ah, so the thought is that the space the groupoidification program anticipated covering is indeed a good space to investigate (it’s not just “idle entertainment”), but that rather than the sketches and hints provided to date, the equivariant stable (co)homotopy approach has shown itself to be more powerful by actually delivering concrete results in this area.
And rather than taking one’s approach as a completely novel way of looking on at matters, the cohomotopy approach can rightfully see itself as growing out of noted predecessors, such as Segal.
I guess that still leaves the question of why one should want to see what part of linear algebra is pure combinatorics. There should be an account which doesn’t go via the needs of M-theory, but for purely mathematical reasons.
Yes, that’s how I feel about it.
I am also eager to eventually hear from Todd about his and Jim Dolan’s secret thoughts on the matter. I am vaguely hoping that they will make the link to something Chow-motive-like as I said in #73, since that would very well harmonize with the structure of spectral Mackey functors in equivariant (co)homotopy: The latter really says that equivariant stable cohomotopy is a motivic kind of thing (as in Kahn-Yamazaki 11, section 2), and my best guess as to what Baez-Hoffnung-Walker is pointing to is a discrete version of something like Chow motives. That might establish the connection. Whatever it is, I’ll be very interested in seeing any details. But I’ll not further ask for them now – promised! :-)
I guess that still leaves the question of why one should want to see what part of linear algebra is pure combinatorics. There should be an account which doesn’t go via the needs of M-theory, but for purely mathematical reasons.
I see what you mean, but my perspective, or at least hope, is different: The idea is to make M-theory grow out of mathematics, so that it becomes a purely mathematical reason itself. But that leads a bit too far here.
Re #80, are we to think of motives within a specific $(\infty, 1)$-topos or as transcending the class of them? I mean, if motives extract the commonality of cohomologies applied to specific entities, one might think of this as operating within a single $(\infty, 1)$-topos via some construction derived from the collection of hom spaces out of an object.
But might one also take motive theory as something more general, something to be expressed in HoTT?
The formulation of genuine $G$-spectra (in any $\infty$-topos) as spectral Mackey functors, hence as spectrally enriched $\infty$-functors on the Burnside category, exhibits them as motives in the sense of “sheaves with transfer” with respect to “spaces” which are just finite $G$-sets, hence a kind of discrete spaces with group actions. (Compare to pure motives or Chow motives or similar, where instead of finite $G$-sets one considers certain schemes.)
Notice how close this is to the intuition advertized in Baez-Hoffnung-Walker: A finite $G$-set may equivalently be thought of as a finite groupoid with isotropy groups contained in $G$, and hence the Burnside category is much like that category of tame spans of finite groupoids. Thus spectral Mackey functors on the Burnside category, and hence genuine $G$-spectra, are much like an $\infty$-category-theoretic realization of the intuition behind what they call the “degroupoidification” functor. (One can also get rid of the explicit dependence on $G$ here by passing further to global equivariant spectra.)
Hi Urs, I’ve begun taking a look at doing the computations you need. Since GAP can produce the table of marks as well as the character table (I have checked this now :-)), it would be nice to do something like in #70 directly (Simon’s code has I think much more than would be needed for the remaining steps).
It would save me some time if you could spell out without the notation how steps 2 and 3 in #70 work. For example, if you could explain how to do these steps in Example 4.1 in your paper, I can probably figure out what is going on.
Actually step 2 is fine, at least for the simplified version. So it is just step 3 that I could do with a bit of help with, due to the notation: is it possible to compute it from the upper triangular matrix?
Thanks, Richard, for starting to look into this!
So for step 3 we need the invertible matrix $U$ which implements the row reduction of step 2 by left multiplication.
The rows of that matrix are the coefficients of the permutation representations $k[G/H_j]$ in the given $V_i$. So the character of $V_i$ is the sum of the characters of $k[G/H_j]$ (which in turn is combinatorially given by number of fixed points) times the corresponding entry of that matrix $U$.
Does that help?
Very helpful! Sorry for the question (am asking in passing, not looked properly), but is there is an easy way to calculate the characters of $G/H_j$? Say just with matrix manipulations of some kind?
Right, so since the character of $k[G/H]$ at $g$ is the number $\vert (G/H)^{\langle g\rangle} \vert$ of fixed points of the cyclic group generated by $g$ on $G/H$, this should be gotten from GAP by calling the table of marks.
Hi Urs, thanks again! I’m getting there. I’ve now implemented the algorithm for getting $\tilde{U}$ and $\tilde{H}$ from the table of marks. But it’s still not entirely clear to me how to get the table on the top of pg.17. For how do we know which rows in the table of marks correspond to conjugacy classes of those subgroups with a single generator? (It seems, both in your paper and in GAP, that the rows of the table of marks are not labelled in such a way as to be able to immediately see this). Apologies if this is completely obvious!
Of course it would not be very difficult to calculate the table of marks from scratch if one knows the conjugacy classes of subgroups. But I don’t know if there’s an easy way to compute what the latter are exactly? Edit: well, it seems that GAP can do it! So, if there is no better way, we can get GAP to compute the subgroup lattice, and get the table of marks from that. But I would imagine it would be slow if the group is of any kind of significant size, unless there are some shortcuts for the kind of groups you’re looking at.
Found some assertions that it is possible to see which conjugacy classes have a cyclic representative just from the table of marks, but have not found a reference for how yet.
Also found that GAP does in some cases allow for finding representatives of the conjugacy classes. For binary dihedral groups I guess computing the subgroup lattice could be done very quickly anyhow.
Have made progress on the implementation. Checked that GAP can get the table of marks in a such a way that the cyclic subgroups can be determined almost instantaneously up to as far as I tried, which was about $Q_{1000}$, so we should have plenty of examples to be able to work with. However, GAP’s table of marks for the binary dihedral group is as follows. A full stop . or nothing means 0.
1: 8
2: 4 4
3: 2 2 2
4: 2 2 . 2
5: 2 2 . . 2
6: 1 1 1 1 1 1
Here 1-5 are cyclic and 6 is not. This is obviously not the same as the ’table of multiplicities’ in your paper/produced by Simon’s code. I imagine that things will still work out and that one should get the same characters for the image of $\beta$ in the end if one applies your algorithm, but I’ve not finished checking this yet.
Although now that I look at Proposition 3.10 in the paper, it looks as though the ’table of multiplicities’ on pg.16 is supposed to be exactly the table of marks, whereas this cannot be the case, because there will always be zero entries in the table of marks. Perhaps you could clarify this before we go further?
Thanks, Richard. Yes, I had mentioned this by email, this was a mistake I had introduced in the v2 document on the nLab. It’s been fixed in the latest version here. The two tables are just not the same (they “sit inside each other”, which had misled me.)
Ah, thanks very much! Do you know if the table of multiplicities is a standard thing considered in the literature? Just wondering for example if it can be computed from the table of marks. Sorry for the naïve questions!
I’ll send you an email on that.
Is it worth remarking somewhere that tables of marks are the $\mathbb{F}_1$-version of character tables? I came across them (#37) by thinking that there should be such a version.
At table of marks Properties it has
The following says that the Burnside character plays the same role for finite G-sets as characters of representations play for finite-dimensional linear representations
Feel invited to expand on that!
Just to say that the answer to Richard’s #95 is now recorded as this Prop.
For the surjectivity of $\beta$ over $\mathbb{Q}$ on $p$-primary groups, I have added pointer also to Theorem 4.4.1 in