This is a question about $\lambda$-rings. A $\lambda$-ring is a commutative ring together with operations $\lambda^n$ for each whole number $n$ which are analogous to the $n$th exterior power and satisfy the same identities. There's the identity $$\lambda^n(x+y)=\sum_{i+j=n}\lambda^i(x)\lambda^j(y)$$ as well as more complicated identities for $\lambda^n(xy)$ and $\lambda^m(\lambda^n(x))$.
My question is: What if instead of exterior powers, we used symmetric ones? As in, we could define an "$s$-ring" to be a ring together with operations $s^n$ satisfying identities analogous to the identities defining a $\lambda$-ring, but for symmetric powers instead of exterior powers. For addition we get $$s^n(x+y)=\sum_{i+j=n}s^i(x)s^j(y),$$ which is the same as before, but the multiplication and iteration identities are different; for instance, instead of $$\lambda^2(xy) = \lambda^1(x)^2 \lambda^2(y) + \lambda^2(x) \lambda^1(y)^2 - 2 \lambda^2(x)\lambda^2(y)$$ and $$ \lambda^2(\lambda^2(x))=\lambda^1(x)\lambda^3(x)-\lambda^4(x)$$ we get $$s^2(xy) = s^1(x)^2 s^1(y)^2 - s^1(x)^2 s^2(y) - s^2(x) s^1(y)^2 + 2 s^2(x)s^2(y)$$ and $$ s^2(s^2(x))=s^2(x)^2 - s^1(x)s^3(x) + s^4(x).$$
(Those are the right analogues, right? I just got those by, in the definitions of $P_k$ and $P_{k,j}$, replacing sets with multisets, and elementary symmetric polynomials with complete homogeneous ones. I didn't actually think about symmetric powers, I just assume that it should work the same.)
My suspicion, based on what I've read, is that this should just be an alternate characterization of a $\lambda$-ring, that it should be possible to express the $s^n$ in terms of the $\lambda^n$ and vice versa. But I don't really know this subject; I'm asking this because I kind of just stumbled across something that might be one of these "$s$-rings".
So: Is this an alternate characterization of a $\lambda$-ring? (If so, could you provide a reference?) And if not, is it a known sort of thing, and where would I read about it?
OK, I now have most of an answer, which I will post here. That said... any references would really be appreciated. I am still looking for references on the matter; I don't want to have to rederive this theory myself.
Anyway, yes, it seems to be an alternate characterization of a $\lambda$-ring, via the usual way of converting between $e_n$ (elementary symmetric polynomials) and $h_n$ (complete homogeneous polynomials), i.e., the relation $\sum_{i=0}^k (-1)^i e_i h_{k-i}=0$, or, in this context, $\sum_{i=0}^k (-1)^i \lambda^i(x) s^{k-i}(x)=0$.
Edit: You could also do the simpler $s^i(x)=(-1)^i \lambda^i(-x)$; this is equivalent by Vandermonde.
I don't quite have this proved, but if I make the probably-true assumption that when verifying these identities in the ring of symmetric functions, or the ring of symmetric functions in two types of variables, that I only need to check elements whose coefficients are nonnegative, then I can prove it. Short version is, ring of symmetric functions is the free $\lambda$-ring on one generator, ring of symmetric functions is the free $\lambda$-ring on two generators (I don't have a reference for that but it seems to follow from the fact that ring of symmetric functions is the free $\lambda$-ring on one generator together with the fact that the elementary symmetric polynomials in two types of variables generate as a ring the ring of symmetric functions in two types of variables), so if it works there it works anywhere. And of course it works there, because that's how the identities are defined (at least, as I said, for the elements without negative coefficients; I am having some trouble handling those).
So I'm not going to accept this answer, because I'm not considering this closed; I'm still looking for a reference so I don't have to rederive the whole theory myself. Thanks all!
Edit: OK, here's a reference that works -- it doesn't explicitly state everything, as such, but it essentially says it. It calls the operations $S^i$, so I guess that's the notation to use. http://lipn.univ-paris13.fr/~duchamp/Books&more/Lascoux/Cbms.pdf I'll accept this answer once the system lets me (unless someone has a better one!); meanwhile I have another, less trivial, question about $\lambda$-rings which I'll ask on MathOverflow soon probably...
Edit: OK, over in chat, Darij Grinberg posted a better reference -- if he wants to post it as a separate answer I'll upvote it, though I can't accept it! -- which is Hazewinkel, Gubareni, and Kirichenko's book "Algebras, Rings, and Modules", volume 3; it discusses lambda rings in a way that puts the $\lambda^i$ and $\sigma^i$ (their terminology) on equal footing. It doesn't explicitly write out the identities as I'd like, but taking a quick look I'm pretty sure they're easily derivable from what they do write. Hooray! This is probably as good as it's going to get.