Question

What do the derived functors of the symmetric powers look like? I understand that this is related to the homology of the symmetric groups, but I don't know a reference for that.

Namely, I'm interested in the homotopy groups of the free simplicial commutative ring on a simplicial set. Let $X_\bullet$ be a simplicial set; I'd like to know the homotopy groups of $\mathbb{Z}[X_\bullet]$. This is the symmetric algebra on the free simplicial abelian group $\mathbb{Z}X_\bullet$, which is weakly equivalent to a product of Eilenberg-MacLane simplicial abelian groups corresponding to the homology of $X_\bullet$ (and is cofibrant). In particular, we have a weak equivalence of simplicial commutative rings $$\mathbb{Z}[X_\bullet] \simeq \bigotimes \mathbb{L} \mathrm{Sym}^\bullet K( H_n(X_\bullet, n)),$$ which brings up the question of what $\mathbb{L} \mathrm{Sym}^\bullet$ looks like. Tyler Lawson points out in answering this question that the answer is somewhat complicated and describes it in low degrees.

Is a complete answer known?

Answer 1

The homology of all of the symmetric groups together is well understood, as Tyler says. Taking mod $p$ coefficients, that is the special case when $X = S^0$ of the calculation of $H_*(CX)$ as a functor of $H_*(X)$, where $C$ is the monad on based spaces associated to any $E_{\infty}$ operad of spaces. The calculation in this form is given as Theorem 4.1, page 40, of [Cohen, Lada, May. The homology of iterated loop spaces, SLN Vol 533. 1976] which is available on my web page. The functor is not all that complicated, but you do have to understand the Dyer-Lashof operations, which are very much like Steenrod operations and can be seen with those as special cases of a general construction of Steenrod operations [A general algebraic approach to Steenrod operations. In SLN Vol. 168. 1970] also on my web page. The paper of Bisson and Joyal cited by Tyler gives a reformulation of this functor in the case $p=2$. If you want the integral homology, that is a mess to write down in closed form, but the mod $p$ Bockstein spectral sequence of $CX$ is entirely determined by that of $X$, as explained in Theorem 4.13 op cit above, so that integral information is also available. It is worth emphasizing that viewing the homology of symmetric groups as a special case of $H_*(CX)$ substantially simplifies both the calculation and understanding the answer.