Let $G$ be an affine groups scheme over $\mathbb Z$. As such it has an associated Hopf algebra, $A=\mathbb Z[G]$ such that $G(R)$ is naturally identified with the set $\hom_{Rng}(A,R)$ of ring homomorphisms, where the group operations (multiplication, inverse, unit) are given on this set from the co-operations of the algebra $A$.

Fix $m\in\mathbb{N}$ and $p$ a prime number, and let $W_m$ be the functor of $p$-typical Witt vectors of length $m+1$. The functor $W_m$ is represenatble as well, with representing algebra $\mathbb{Z}[x_0,\ldots,x_m]$.

I am interested in the structure of the group scheme $R\mapsto G(W_m(R))$. Greenberg's results imply that this functor is a an affine group scheme as well, and hence representable. Is there any known construction for the associated Hopf-algebra of $G\circ W_m$?

Some obvious facts-

In the case where the prime $p$ is invertible in $R$, the Witt ring $W_m(R)$ is isomorphic to the product ring $\prod_{i=0}^m R$. Since the group scheme $R\mapsto G(\prod_{i=0}^{m} R)=\prod_{i=0}^{m}G( R)$ is represented by the Hopf algebra $A^{\otimes m+1}=\underbrace{A\otimes\cdots\otimes A}_{m+1\text{-fold}}$, I somehow expect there to exist a map between the representing algebra of $G\circ W_m$ and $A^{\otimes m+1}$, which becomes an isomorphism under localization by $p$.

On the other hand, in the complementary case, the group $G(W_m(R))$ is usually nothing like $G(R)\times G(R)$. For example, in the case $G= GL_n$, $R=\mathbb{F}_p$ and $m=2$, we have an exact sequence $$1\to M_n(R)\to GL_n(W_2(R))\to GL_n(R)\to 1.$$ In particular, $|GL_n(W_2(R))|\ne |GL_n(R)\times GL_n(R)|$, and one cannot expect any sort of bijection to exist between the two.

I'm new to the subject, and my foundations on algebraic groups or algebraic geometry are not incredibly solid, so I apologize if anything I wrote above does not make complete sense, or is obviously false.

I would very much appreciate any clue or reference to the construction of the Hopf algebra of $G\circ W_m$, or any other interesting facts regarding the structure of $G\circ W_m$.

Thank you very much! Shai

Plethystic algebra: The coordinate Hopf algebra of $G \circ W_m$ will be the biring $S_m \odot A$, where $S_m$ is the coordinate Hopf algebra of $W_m$ (that is, $\mathbb{Z}\left[x_0,\ldots,x_m\right]$ with an appropriate coalgebra structure). This is just abstract nonsense. The question is, of course, how much we can say about this biring. $\endgroup$ – darij grinberg Dec 31 '17 at 12:24Pseudo-reductive Groups(the assumption there that $k=\overline{k}$ is not needed for this), where a harder fact is also shown: for non-commutative connected reductive $G$, $G\circ W_2$ has no Levi $k$-subgroup! $\endgroup$ – nfdc23 Dec 31 '17 at 13:04