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Let $\Sigma_k$ be the permutation group of order $k$.

Let $r: \Sigma_k\to GL(k)$ be the regular representation by permuting the order of the standard basis of $\mathbb{R}^n$.

Let $\rho: B\Sigma_k\to BGL(k)=G_k(\mathbb{R}^\infty)$ be the induced map (induced from $r$) between classifying spaces.

Let $\rho^*: H^*(G_k(\mathbb{R}^\infty))\to H^*(B\Sigma_k)$ be the induced homomorphism of cohomology with coefficients in $\mathbb{Z}_2$.

What is the image $\text{Im} \rho^*$?

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  • $\begingroup$ This paper by Pierre Guillot contains relevant information. $\endgroup$ – few_reps Oct 20 '14 at 11:19
  • $\begingroup$ @few_reps Guillot considers the subring generated by all real representations, whereas the original post is concerned with the subring generated by one particular representation. $\endgroup$ – user43326 Oct 20 '14 at 16:03
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I suppose by $\mathbb{Z}_2$ you mean $\mathbb{Z}/2$ and not $\mathbb{Z}^{\wedge}_2$. Then we have $$H^*(BO(k);\mathbb{Z}/2)\cong \mathbb{Z}/2[w_1,w_2\cdots w_k].$$ By definition we have $\rho ^*(w_n)=w_n(r)$ where $w_n(r)$ is the $n$-th Stiefel-Whiteney class of the representation $r$. Since the regular representation factors through $O(k-1)$, we see that $w_k=0$. We can, in theory, determine the image using the fact that the Quillen's map is injective (as a matter of fact, isomorphic) for the symmetric groups, that is elements of mod $2$ cohomology of the symmetric get detected by its elementary abelian subgroups.

In the case of $k=4$, for example, if we call $V_1=\{(12),(34),(12)(34),id\},V_2=\{(12)(34),(13)(24),(14)(23),id\}$ then the restriction of $w_n(r)$'s to $BV_1$ can be computed using Whitney product formula, and the restriction to $BV_2$ can be found more or less using the definition of Dickson invariants, so by comaparing with the table in http://homepages.math.uic.edu/~bshipley/ConMcohomology1.pdf Example 4.4, we see that they coincide with standard generators of $H^*(B\Sigma _4;\mathbb{Z}/2)$. Thus for $k=4$ $\rho ^*$ is surjective. In principle, this kind of analysis can be carried out for larger $k$, but I am not aware of any explicit result.

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