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I added a Definition section to Burnside ring (and made Burnside rig redirect to it).
And what can be said about the relationship to the Burnside category? Wikipedia has it as a categorification.
Also, Burnside rigs can be defined more generally. e.g., for distributive categories.
I wasn’t aware of Schanuel’s terminology – thanks.
I’m curious when the Burnside ring coincides with the representation ring (say working over the ground field $\mathbb{C}$). Jim Dolan once showed me some fairly convincing empirical evidence that the Burnside ring of the symmetric group $S_n$ is the same as the representation ring; an example is given at the page Gram-Schmidt process. But I never seriously attempted to find a proof.
added more and original references for the statement that the Burnside ring is isomorphic to the equivariant stable cohomotopy of the point
$A(G) \simeq \mathbb{S}_G(\ast)$Am adding this also at equivariant stable cohomotopy
The canonical map of spectra $\mathbb{S} \longrightarrow KU$ becomes, when restricted to the point with a $G$-action, the ring homomorphism from the Burnside ring $A(G)$ to the representation ring $R(G)$
$A(G) \longrightarrow R(G)$which sends a finite $G$-set $S$ to the vector space spanned by $S$ and equipped with the induced linear permutation representation.
This is just the kind of map that underlies Dolan-Baez’s old “groupoidification” idea. There the goal was (I suppose) to see how much of the linear algebra can be understood to arise from the combinarics somehow in a conceptual way.
Here one might want to ask a similar question: How much of $KU$ is “induced” by $\mathbb{S}$? Or to start with, how much of $R(G)$ is induced by $A(G)$? Is the former something we’d obtain from the latter by following some god-given path?
Here one might want to ask a similar question: How much of $KU$ is “induced” by $\mathbb{S}$? Or to start with, how much of $R(G)$ is induced by $A(G)$? Is the former something we’d obtain from the latter by following some god-given path?
There are related questions asked at Gram-Schmidt process, where the toy example is of representations of symmetric groups. When I saw Jim Dolan and Simon Burton back in February, we were musing about the coincidence of the number of Young diagrams being the same as the number of conjugacy classes, and I thought I heard Jim suggesting that the Young diagrams might be better understood in terms of conjugacy classes of a (Langlands) dual.
Anyway, I’d like to understand this myself.
Thanks for the pointer!
Oh, wow, am I reading this correctly as directly implying that the homomorphsim
$A(S_n) \longrightarrow R(S_n)$is in fact surjective?!
Re #6, I had a similar idea back here after David Ben-Zvi told us about Langlands duality:
A thought after all these months: if symmetric groups fall into this picture by being $GL(n,\mathbb{F}_1)$, and if GLs are Langlands self-dual, this explains how Young diagrams parameterize conjugacy classes and at the same time irreducible representations.
Interesting, thanks.
Next I’d like to know how much of all this survives changing the group to a finite subgroup of $SU(2)$. But need to run now…
Re #7: that’s what I had understood, but I certainly don’t have a proof for general $n$.
There may be a more categorified way of saying this, considering a suitable category of “virtual permutation representations”, where the map $A(S_n) \to R(S_n)$ is promoted to an essentially surjective functor, but I’d need to think more how I’d like to say it.
I see, thanks.
How about taking $G = A_4$ the alternating group. Might the construction still give a surjection $A(A_4) \to R(A_4)$?
Actually, I would be interested in knowing it for the “binary alternarting group” $2 A_4$, i.e. the “double cover” of the tetrahedral group.
Really I’d like to know it for all the finite subgroups of $SU(2)$, but here I am thinking that maybe the (binary) alternating ones are close enough to the symmetric groups that it would be easy to adopt the proof.(?)
Urs, I believe Simon Burton may have carried out calculations of the categorified Gram-Schmidt process for groups like that (we were discussing the dihedral group of order $8$). I’ll ask him.
For what it’s worth, this article has the details for the Burnside ring of the icosahedral group.
OK, even better.
This page says that the map from the Burnside ring to the representation ring for $S_4$ is not injective.
In fact, the page here says that the comparison map is only surjective for the trivial group and any two-element group, and is “seldom” injective.
EDIT: one last one. In Burnside’s book, on pdf page 272, there the ’table of marks’ for $A_4$ and what seems to be at least part of the multiplication structure of the Burnside ring.
EDIT again: tom Dieck gives information about $A_5$.
If I find something about the binary versions I will link to it.
Thanks for all the pointers! Am reading…
But now I am confused: Isn’t Todd’s discussion at “Categorified Gram-Schmidt process” saying that $\beta$ is surjective, for $S_4$?
But your first link, at the very bottom, confirms Todd’s computation. Clearly this makes beta surjective, or else I am missing something basic.
There is
which seems to discuss the surjectivity, or not, head on.
But need to run now.
Hmm, a mystery indeed! I’m not sure what’s going on. I agree that it looks like the formulas at the bottom of this page show that every basis vector in the representation ring $R(S_4)$ is in the image of $\beta$.
Worse, PlanetMath cites a theorem of Segal that says for $p$-groups, and the representation ring over $\mathbb{Q}$, $\beta$ is always surjective! (and an isomorphism for such groups iff they are cyclic)
Thanks!
So it’s that one page www.maths.manchester.ac.uk/~jm/wiki/Representations/Burnside which disagrees with all other sources that we have seen so far, on the surjectivity of $\beta$. Might it be that the error there is caused by the evident glitch of not thinking about virtual representations, but plain representations?
Now that I am back on my laptop machine:
Indeed, the article from #18
has detailed references regarding surjectivity of $\beta$ or not: see the paragraph in the middle of the first page.
In particular they cite what must be Segal’s result that PlanetMath is alluding to (at planetmath.org/RepresentationRingVsBurnsideRing), but together with a bunch more.
PlanetMath also says that $\beta$ is an iso for all cyclic groups! This is most interesting in view of my quest for $\beta$ in the case of ADE groups. But I haven’t tracked down a real reference for this statement yet.
(To be sure, in the nLab I was considering the ground field to be $\mathbb{C}$ or an algebraically closed field.)
Oh, the statement that $\beta$ is surjective for all cyclic groups follows from the statement for $p$-groups from the fundamental theorem of cyclic groups! Right?
I am starting a proposition collecting the known surjectivity results here.
To be expanded and improved (need to add assumptions on ground field. For the moment everything is in char 0, I suppose.)
have briefly been trying to find (in the literature, that is :-) the proof that $\beta$ is an iso for cyclic groups. From the way this is stated on PlanetMath (here) I gather this is meant to be evident from the proof that Segal gives of surjectivity, or else from the formulas in Hambleton-Taylor 99, but if so, I need to spend more time with it.
just discovered that Ben Webster gave the simple argument why $\beta$ is injective precisely for the cyclic groups, here!
Together with the surjectivity from Segal’s theorem, this shows that $\beta$ is an isomorphism for cyclic groups. Probably the author of that PlanetMath entry found the injectivity argument too trivial to mention here.
[ have added that to the entry, in this prop. ]
Okay, so I have now much of what I was after, but only for rational representations. Need to think about how much of this carries over to complex representations…
Oh, I see now.
$\mathbb{Q}$ is a splitting field only for $\mathbb{Z}/2$, but for none of the other cyclic groups (here). Hence surjectivity of $\beta$ over $\mathbb{Q}$ breaks after passage to $\mathbb{C}$ for all $\mathbb{Z}/n$ with $n \gt 2$.
I suppose this resolves the apparent contradiction above in #20 ! (Maybe that’s what Todd was trying to tell us in #22. Sorry for being slow.)
I have emailed James Montaldi on the issue with the webpage www.maths.manchester.ac.uk/~jm/wiki/Representations/Burnside mentioned around #20 above. He agreed that this was in error and has removed the statement about surjectivity now.
I do not know if it helps, but I had colleagues who used to work on structures related to lambda rings on the Burnside ring and looking up lambda rings I find a reference to James Borger’s paper on Lambda rings and the field with one element. The Burnside ring does have a pre-$\lambda$ -ring structure if I remember rightly, but not a ‘special’ one.
One of the papers was ‘Adams operations and λ-operations in β-rings by I. Morris and C.D. Wensley, another is Computing Adams operations on the Burnside ring of a finite group. J. Reine Angew. Math. 341 (1983), 87–97, by G. Morris and the other two authors.
These are mentioned in this MO question. This relates to a conjecture in Knutson, Donald (1973), λ-rings and the representation theory of the symmetric group, Lecture Notes in Mathematics, 308, which dies nit seem o quite make sense.
Does this relate to the problem of the surjectivity etc. as the representation ring is a Lambda ring?
The point in Guillot 06 “Adams operations in cohomotopy” is to show that it is not quite a $\lambda$-ring, but a “$\beta$-ring”.
I believe the initial notion of $\beta$-ring is, in fact, due to another ex-colleague! (Remember Dudley Littlewood was Ronnie Brown’s predecessor at Bangor and he was central in the development of permutation representation theory.) I really should look back over that stuff, and will if I have a moment. Guillot’s paper looks interesting. Thanks.
added remark that Segal’s theorem $A(G) \overset{\simeq}{\longrightarrow} \mathbb{S}_G(\ast)$ is a special case of the tom Dieck splitting theorem
There might be some interesting ideas about $G$-equivariant $\mathbb{F}_1$-theory in
Thanks for the pointer.
He attributes the observation that the Barratt–Priddy–Quillen theorem may be read as sying $\mathbb{S} \simeq K \mathbb{F}_1$ to
I have looked through that pdf. While I see it talk about $\mathbb{F}_1$, I didn’t find a remark yet concerning Barratt–Priddy–Quillen…
added pointer to
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