For any ring R, $\bigsqcup_n {BGL}_n(R)$ is an $E_\infty$-space. Are there examples of rings where people have calculated $H_*(\bigsqcup_n {BGL}_n(R);\mathbb{Z}/2)$ and determined the Dyer-Lashof operations?
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1$\begingroup$ Do you necessarily want $\amalg_n BGL_n(R)$ or would you be ok with the group completion (the homology is not that different)? It seems to me that it would be easier to look for results on the homology of algebraic K-theory. $\endgroup$– Denis NardinApr 29, 2015 at 20:42
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Taking the ring of real numbers $\mathbb{R}$, complex numbers $\mathbb{C}$, or $\mathbb{H}$, the paper of Priddy
`DYER-LASHOF OPERATIONS FOR THE CLASSIFYING SPACES OF CERTAIN MATRIX GROUPS' Quart J. Math. Oxford (3), 26 (1975), 179-93
provides example of computations and detailed formula of the sort you may look for, bearing in mind that in these cases the monoid you take, is homotopy equivalent to the one considered in the paper.
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3$\begingroup$ Here is a complementary and earlier reference: Stanley O. Kochman. Homology of the classical groups over the Dyer-Lashof algebra. Trans. Amer. Math. Soc. 185 (1973), 83-136. $\endgroup$ Apr 30, 2015 at 1:44