# Is the Burnside ring a lambda-ring? + conjecture in Knutson p. 113

Warning: I'll be using the "pre-$\lambda$-ring" and "$\lambda$-ring" nomenclature, as opposed to the "$\lambda$-ring" and "special $\lambda$-ring" one (although I just used the latter a few days ago on MO). It's mainly because both sources use it, and I am (by reading them) slowly getting used to it.

Let $G$ be a finite group. The Burnside ring $B\left(G\right)$ is defined as the Grothendieck ring of the category of finite $G$-sets, with multiplication defined by cartesian product (with diagonal structure, or at least I have difficulties imagining any other $G$-set structure on it; please correct me if I am wrong).

For every $n\in\mathbb{N}$, we can define a map $\sigma^n:B\left(G\right)\to B\left(G\right)$ as follows: Whenever $U$ is a $G$-set, we let $\sigma^n U$ be the set of all multisets of size $n$ consisting of elements from $U$. The $G$-set structure on $\sigma^n U$ is what programmers call "map": an element $g\in G$ is applied by applying it to each element of the multiset. This way we have defined $\sigma^n U$ for every $G$-set $U$; we extend the map $\sigma^n$ to all of $B\left(G\right)$ (including "virtual" $G$-sets) by forcing the rule

$\displaystyle \sigma^i\left(u+v\right)=\sum_{k=0}^i\sigma^k\left(u\right)\sigma^{i-k}\left(v\right)$ for all $u,v\in B\left(G\right)$.

Ah, and $\sigma^0$ should be identically $1$, and $\sigma^1=\mathrm{id}$. Anyway, this works, and gives a "pre-$\sigma$-ring structure", which is basically the same as a pre-$\lambda$-ring structure, with $\lambda^i$ denoted by $\sigma^i$. Now, we turn this pre-$\sigma$-ring into a pre-$\lambda$-ring by defining maps $\lambda^i:B\left(G\right)\to B\left(G\right)$ by

$\displaystyle \sum_{n=0}^{\infty}\sigma^n\left(u\right)T^n\cdot\sum_{n=0}^{\infty}\left(-1\right)^n\lambda^n\left(u\right)T^n=1$ in $B\left(G\right)\left[\left[T\right]\right]$ for every $u\in B\left(G\right)$.

Now, let me quote two sources:

Donald Knutson, $\lambda$-Rings and the Representation Theory of the Symmetric Group, 1973, p. 107: "The fact that $B\left(G\right)$ is a $\lambda$-ring and not just a pre-$\lambda$-ring - i. e., the truth of all the identities - follows from [...]"

Michiel Hazewinkel, Witt vectors, part 1, 19.46: "It seems clear from [370] that there is no good way to define a $\lambda$-ring structure on Burnside rings, see also [158]. There are (at least) two different choices giving pre-$\lambda$-rings but neither is guaranteed to yield a $\lambda$-ring. Of the two the symmetric power construction seems to work best." (No, I don't have access to any of these references.)

For a long time I found Knutson's assertion self-evident (even without having read that far in Knutson). Now I tend to believe Hazewinkel's position more, particularly as I am unable to verify one of the relations required for a pre-$\lambda$-ring to be a $\lambda$-ring:

$\lambda^2\left(uv\right)=\left(\lambda^1\left(u\right)\right)^2\lambda^2\left(v\right)+\left(\lambda^1\left(v\right)\right)^2\lambda^2\left(u\right)-2\lambda^2\left(u\right)\lambda^2\left(v\right)$ for $B\left(G\right)$.

What also bothers me is Knutson's "conjecture" on p. 113, which states that the canonical (Burnside) map $B\left(G\right)\to SCF\left(G\right)$ is a $\lambda$-homomorphism, where $SCF\left(G\right)$ denotes the $\lambda$-ring of super characters on $G$, with the $\lambda$-structure defined via the Adams operations $\Psi^n\left(\varphi\left(H\right)\right)=\varphi\left(H^n\right)$ (I think he wanted to say $\left(\Psi^n\left(\varphi\right)\right)\left(H\right)=\varphi\left(H^n\right)$ instead) for every subgroup $H$ of $G$, where $H^n$ means the subgroup of $G$ generated by the $n$-th powers of elements of $H$. This seems wrong to me for $n=2$ and $H=\left(\mathbb Z / 2\mathbb Z\right)^2$ already. And if the ring $B\left(G\right)$ is not a $\lambda$-ring, then this conjecture is wrong anyway (since the map $B\left(G\right)\to SCF\left(G\right)$ is injective).

Can anyone clear up this mess? I am really confused...

Thanks a lot.

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Your LaTeX in the last paragraph is sticking out into the "Related" section, or at least it appears this way to me (I'm using Chrome). You may want to put that long bit on a separate line. – Zev Chonoles Feb 5 '10 at 23:43
Hmm, I don't see much latex in my last paragraph. Maybe you mean the $\lambda^2\left(uv\right)$ formula? Okay, will put it in a new line. – darij grinberg Feb 6 '10 at 0:20
Your equation defining lambda^n in terms of sigma^n doesn't look right. I'm sure there needs to be a minus sign with the t in the series for the lambda's. – Charles Rezk Feb 6 '10 at 1:21
You're right, thanks. – darij grinberg Feb 6 '10 at 9:44

On p. 90, at the end of section 2, they say: "Knutson conjectured that the Adams operations on SCF(G) inherited from A(G) [=Burnside ring of G] are given by [the formula you mentioned, involving the subgroup generated by nth powers of a subgroup $K$]. We will show that this is correct if $K$ is cyclic, but not true in general."
Thanks a lot. This kills the conjecture at least. I assume that for cyclic $K$, it is not particularly hard (one can wlog assume that $G=K$, and $B\left(G\right)$ for cyclic $G$ should be some kind of stunted Witt rings). – darij grinberg Feb 6 '10 at 9:39
Okay, I now found the first page of the reference: reference-global.com/doi/abs/10.1515/crll.1983.341.87 and indeed it claims that the Burnside ring is just pre-$\lambda$ rather than $\lambda$. And a reference to Siebeneicher claiming that it is $\lambda$ if $G$ is cyclic (which should be Witt vector theory again). My question is settled. – darij grinberg Feb 6 '10 at 10:49