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](http://en.wikipedia.org/wiki/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](http://arxiv.org/abs/0804.3888)*, 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.



Is the Burnside ring a lambda-ring? + conjecture in Knutson p. 113
I've just gone and looked up [158] (Gay, C. D.; Morris, G. C.; Morris, I.
Computing Adams operations on the Burnside ring of a finite group.
J. Reine Angew. Math. 341 (1983), 87--97.


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."


I haven't looked at it carefully, but they appear to give some more complicated looking formulas for the action of Adams operations on super-characters, valid in some cases.


They don't seem to mention Knutson's claim that the Burnside ring is a lambda-ring (not merely pre-lambda). 


