As the title says, I would like to know what the fundamental theorem of algebraic K-theory would say over the field with one element. Recall that the fundamental theorem of K-theory provides a decomposition $K\_i(R[T,T^{-1}])\cong K\_i(R)\oplus K\_{i-1}(R)$ for $R$ regular. 


Now turn to the philosophical part, the field $\mathbb{F}\_1$. 
For a root system $\Phi$, the natural candidate for the
algebraic group of type $\Phi$ over $\mathbb{F}\_1$ is
the Weyl group $W(\Phi)$ - the formula for the group order is right as
noted by Tits, see also several other MO questions on
$\mathbb{F}\_1$ such as [this](https://mathoverflow.net/q/272498/10446) or [this](https://mathoverflow.net/questions/77507/braid-groups-and-gl-n-mathbbf-1t). Now I want to understand the loop group of this group over $\mathbb{F}\_1$. The notation would probably be something like $G(\Phi,\mathbb{F}\_1[T,T^{-1}])$ and I
apologize if anyone is offended by this abuse of notation. I
would expect the affine Weyl group for the affine root system
$\tilde{\Phi}$ to be a reasonable candidate. 


The above philosophical guessing has some consequences, and I would like to know if the following statements have been considered, and if they are true or false. In the following, I specialize to the case of root systems $A\_n$ and $\tilde{A}\_n$, just for definiteness. In that case $W(A\_n)\cong S\_{n+1}$ is the symmetric group,
and the affine Weyl group $W(\tilde{A}\_n)$ is an extension of $S\_{n+1}$
by $\mathbb{Z}^{n}$, where $\mathbb{Z}^n$ is identified with the hyperplane $x\_1+\cdots+x\_{n+1}=0$ in $\mathbb{Z}^{n+1}$ with the permutation action of $S\_{n+1}$. Now here are the questions:


1. Are there stabilization theorems for the homology of
$W(\tilde{A}\_n)$? How do
these stabilization theorems compare to those for $S\_{n+1}$, in particular
is the stable range the same or one less? More generally, has the
homology of $W(\tilde{A}\_n)$ been computed by any chance (this is
loosely related to [this MO-question](https://mathoverflow.net/questions/95860/second-cohomology-of-group-of-s-n) on cohomology of the action of $S\_n$ on $T\_n$).
2. Provided there are stabilization theorems, have people considered the
group completion of $\bigsqcup\_n BW(\tilde{A}\_n)$ and compared it to the
plus-construction space $BW(\tilde{A}\_\infty)^+$? A stabilization theorem should imply that the two are weakly equivalent.
3. Does the fundamental theorem of algebraic K-theory hold in this
situation? In other words, can we express the homotopy groups of
$BW(\tilde{A}\_\infty)^+$ in terms of stable homotopy groups of
spheres? Maybe $BW(\tilde{A}\_\infty)^+$ even splits as two copies of the
sphere spectrum (one copy suitably shifted)?


Of course, similar questions can be formulated for the other classical
series, and I would be happy to know the answers in these cases as well.



Fundamental theorem of K-theory for loop groups over $\mathbb{F}_1$?
Let $G\_n := W(\tilde{A}\_{n-1})$. If I understand your description correctly, there is an extension
$$1 \to G\_n \to S\_{n} \wr \mathbb{Z} \overset{sum}\to \mathbb{Z} \to 1$$
and so a $\mathbb{Z}$-Galois cover $BG\_n \to B(S\_{n} \wr \mathbb{Z})$. There are exterior products
$$\mu\_{n,m} : G\_n \times G\_m \to G\_{n+m}$$
given by concatenation, and these make $M := \coprod\_{n \geq 0} BG\_n$ into a homotopy commutative topological monoid. There are similar maps making $M' := \coprod\_{n \geq 0} B(S\_{n} \wr \mathbb{Z})$ into a homotopy commutative topologial monoid,


By the group-completion theorem of McDuff--Segal, the associated maps
$$\mathbb{Z} \times BG\_\infty \longrightarrow \Omega B M \quad \quad \mathbb{Z} \times B(S\_\infty \wr \mathbb{Z}) \longrightarrow \Omega B M'$$
are both homology equivalences. In fact, by an addendum to the McDuff--Segal work in my paper "Group-Completion", local coefficient systems, and perfection, Quarterly Journal of Mathematics 64 (3) (2013) 795-803, these maps are both in fact acyclic, so yield homotopy equivalences
$$\mathbb{Z} \times BG\_\infty^+ \simeq\Omega B M \quad \quad \mathbb{Z} \times B(S\_\infty \wr \mathbb{Z})^+ \simeq \Omega B M'.$$


The space $B(S\_n \wr \mathbb{Z})$ can be modelled as configurations of $n$ points in $\mathbb{R}^\infty$ with labels in $B\mathbb{Z} = S^1$. As such, the group completion $\Omega B M'$ can be approached using Segal's "scanning" technique, and there is a homotopy equivalence
$$\Omega BM' \simeq Q(B\mathbb{Z}\_+)$$
to the infinite loop space of the spectrum $\mathbf{S}^0 \vee \mathbf{S}^1$.


To get at $\Omega BM$, we can use that the fibration sequence
$$BG\_\infty \to B(S\_\infty \wr \mathbb{Z}) \to B\mathbb{Z}$$
satisfies a condition (e.g. Berrick's) to be plus-constructible, so there is a fibration sequence
$$\Omega BM \to Q(B\mathbb{Z}\_+) \to B\mathbb{Z}.$$
Thus $\Omega BM$ is the infinite loop space of the spectrum $\mathbf{S}^0 \vee \overline{\mathbf{S}}^1$, where $\overline{\mathbf{S}}^1$ is the 1-connected cover of $\mathbf{S}^1$.


I have carefully avoided discussion of homological stability for the groups $G\_n$ (it is not, despite what the question says, necessary (or sufficient) to compare $\mathbb{Z} \times BG\_\infty^+$ with $\Omega BM$). I think that homological stability does hold, but it requires a technique not yet in the literature. This technique will hopefully appear in a forthcoming paper.


