I am reading now Tyler Lawson's *$E_n$ ring spectra and Dyer-Lashof operations* form the *Handbook of Homotopy Theory* and I've got a question on the Remark 1.4.19.

We have an operad $\mathcal{O}$ and $\Sigma_k$ acts freely and properly discontinuosly on $\mathcal{O}(k)$. Let $V\subset\mathbb{R}^k$ consisting of elements summing up to $0$ and let $\rho\to B\Sigma_k$ be associated vector bundle of dimension $k-1$ (1).

Now we can form for any $m$ an associated vector bundle $\mathbb{R}^m\otimes\rho$. If we define $P(k)$ as $\mathcal{O}(k)/\Sigma_k$, then there is a virtual bundle $m\rho$ on $P(k)$. Then the Thom spectrum $P(k)^{m\rho}$ is canonically equivalent to the spectrum $\Sigma^{-m}\Sigma^{\infty}_+\mathcal{O}(k)\otimes_{\Sigma_k}(S^m)^k$, where the latter appears in the definition of power operations (2).

So my question(s) are:

What is the associated vector bundle $\rho$ appearing in (1)? Is it just subbundle of the trivial bundle?

How do we get the equivalence in (2)?