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For the symmetric group $S_3$, it is classically known that \begin{equation} H^*(S_3;\mathbb{Z})\cong \mathbb{Z}[x,y]/(2x,6y,x^2-3y), \end{equation} where $|x|=2$ and $|y|=4$. Moreover, $x$ can be identified with the first (and top) Chern class of the 1-dimensional sign representation of $S_3$, and likewise $y$ can be identified with the top Chern class of the standard representation.

Question 1: Is something similar true for the integral cohomology of $S_4$? Namely, is there an explicitly computed presentation of $H^*(S_4;\mathbb{Z})$ whose generators $x,y,z$ are the respective top Chern classes of (1) the sign representation, (2) the 2-dimensional representation $S_4$ given by composing the projection $S_4\rightarrow S_3$ with the standard representation of $S_3$, and (3) the standard representation of $S_4$?

Question 2: Is there an analogous situation for $H^*(S_4; \mathbb{Z}_2)$ in terms of Stiefel-Whitney classes of the corresponding real representations?

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For Q2, my collaborators and I show that all mod-two cohomology of symmetric groups is generated by Stiefel-Whitney classes of standard representations, if you allow both cup product and transfer (induction) product. It is better, however, to take other Hopf ring generators, in which case all cohomology of symmetric groups is a free divided powers Hopf ring on classes in $H^{2^k - 1}(S_{2^k})$. See https://arxiv.org/abs/0909.3292, in particular Section 10 for discussion of Stiefel-Whitney classes.

For Q1, the integral groups are known, but not ring structure. (I happen to be working out the Bocksteins now, at the prime two.) I haven't considered the characteristic class question.

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    $\begingroup$ The integral cohomology ring of $S_4$ is known. It's given in the paper of Soule´ on the integral cohomology ring of $SL_3(\mathbb{Z})$. $\endgroup$
    – tj_
    Commented Jul 3, 2019 at 22:47
  • $\begingroup$ For completeness, Soulé's Proposition 2.iii gives the 2-primary component quotient ring (of the full ring) explicitly, while his Lemma 7 gives the 3-primary quotient ring indirectly in terms of that of $S_3$ (which we can then look up elsewhere), and then we take the product. $\endgroup$
    – Chris Gerig
    Commented Dec 5, 2019 at 4:36
  • $\begingroup$ Is anything known about $H^*(S_4,\mathbb{Z}_4)$? $\endgroup$
    – Noah B
    Commented Sep 25, 2023 at 22:21
  • $\begingroup$ Yes. First, I should have elaborated that the higher torsion is all determined by a result treated very briefly on pages 48-49 of Cohen-Lada-May's book. What I was working on before, but have paused that project, is a conjecture that in cohomology higher torsion all arises from divided powers operations. For $S_4$ this means that there is 4-torsion in degrees which are multiples of 4, powers of $y^2$ using notation from the question above. (E-mail me if you want to know more.) $\endgroup$
    – Dev Sinha
    Commented Sep 26, 2023 at 23:07

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