Does anybody know a reference for the explicit description of the integral cohomology ring of $S_5$ and $S_6$. I can not find them anywhere in the internet. For $S_4$, I found C. B. Thomas's nice article published in Mathematika, in 1974.
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1$\begingroup$ Salvetti computes the integral cohomology of $S_5$ (but not the ring structure) in "Cohomology of Coxeter groups", Topology Appl. 118 (2002), no. 1-2, 199–208. The main point for him is that he can compute the cohomology with coefficients in any Specht module. One can use the Salvetti complex to compute the cohomology of any Coxeter group. See his article with de Concini. I know you are interested in the ring structure, but in case you did not know the additive structure, this is a place to start. $\endgroup$– Daniel JuteauCommented Mar 5, 2015 at 22:37
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1$\begingroup$ Oh I just found a related question (again, this is only additively...): mathoverflow.net/questions/180637/… $\endgroup$– Daniel JuteauCommented Mar 5, 2015 at 23:19
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$\begingroup$ Thanks for the comments. I had just found Leon's question today too. It is upto dimension 10. $\endgroup$– Bob DobbsCommented Mar 6, 2015 at 4:03
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$\begingroup$ Not what you asked, but the mod-2 cohomology ring of $S_6$ was computed by P.Petek [Glasnik Mat. Ser. III 12(32) (1977), no. 2, 247–258]. The paper is available on Google books books.google.com/books?id=Ff2V3qDC32IC&pg=PA247 $\endgroup$– Dave Witte MorrisCommented Mar 6, 2015 at 18:46
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