Surjectivity of representations in algebraic K-theory

Question

Let $G$ be a finite group with finite-dimensional irreducible representations $\rho_i:G\to\mathrm{GL}_{n_i}(k)$ over a field $k$ indexed by $i=1,...,m$. These compose with the canonical map $\mathrm{GL}_{n_i}(k)\to\mathrm{GL}(k)$ to give $\rho_i:G\to\mathrm{GL}(k)$ for $i=1,\cdots,m$. Taking classifying spaces and applying the Quillen plus construction therefore gives maps $\rho_i:BG^+\to B\mathrm{GL}(k)^+$, and hence homomorphisms $\rho_i^n:\pi_n BG^+\to K_n(k)$ for every $1,\cdots,m$, where $K_n(k)$ denotes the $n$th algebraic K-theory group of $k$. Have these maps been studied somewhere? When are these maps surjective?

An easy example is that $\rho_i^{2n}=0$ for $n>0$ if $k=\mathbf{F}_p$. In general, though, I believe studying these maps might be hard, because every path connected space arises as $BG^+$ for some discrete group $G$ (Kan-Thurston). As a special case, what can one say about these maps if $\mathrm{char}(k)=p>0$ and $G$ is a $p$-group?

Answer 1

I don't actually know places in the literature where induced maps from representations have been studied (except from the Brauer lift used in Quillen's computation of K-theory of finite fields). The following are some remarks and observations which partially answer the question.

As a first remark, I assume that the plus-construction applied to $BG$ is the one for the maximal perfect subgroup. If the maximal perfect subgroup is trivial, then $BG\to BG^+$ is a homotopy equivalence and then $\rho^n=0$ for $n>1$. This happens e.g. for abelian groups. It also happens for $p$-groups, since any $p$-group strictly contains its commutator subgroup and therefore the maximal perfect subgroup must be trivial (which answers the last question).

Although the Kan-Thurston theorem allows to write any path-connected space as $BG^+$ for some discrete group, this group will usually be infinite. If you allow infinite groups, then $G=GL(k)$ will give a surjective map trivially. For $G$ finite, there are strong restrictions on the homotopy type of $BG^+$. Some information on the homotopy groups of $BG^+$ for finite groups can be found in

• R. Levi: On finite groups and homotopy theory. Mem. Amer. Math. Soc. 118, 1995.

• F.R. Cohen and R. Levi: On the homotopy theory of $p$-completed classifying spaces. In: Group representations: cohomology, group actions and topology. Proc. Sympos. Pure Math. 63, Amer. Math. Soc., 1998.

The homotopy groups of $BG^+$ for finite perfect groups are torsion, so the maps $\rho^n$ can only be surjective if the K-groups of the field $k$ are finite which will probably be very rare. For $k=\mathbb{F}_q$, one can of course use $GL_n(\mathbb{F}_q)$ to get surjections on K-groups in degrees below $n$ (by the stabilization theorem).

In general, it would seem more appropriate to ask if a particular torsion class lies in the image of a map $\rho^n$ for some finite group. For algebraically closed fields of characteristic $p$, all the torsion classes lie in the image of some representation, this follows from the above observation for $k=\mathbb{F}_q$ and Suslin's rigidity theorem. Maybe the Brauer lift used by Quillen for the computation of K-theory of finite fields allows to show that torsion in the algebraic K-theory of $\mathbb{C}$ can be represented using representations of finite groups.