As explained in this question , there are two $\lambda$-ring structures on ${\mathbb Z}[x]$. In layman's terms, both come from a realization of ${\mathbb Z}[x]$ as the representation ring of an algebraic (semi)-group: it is either $R({\mathbb C},\times)$, or $R(SL_2({\mathbb C}))$.

Question 1: What are $\lambda$-ring homomorphisms between these? As we can choose any of the two structures for the range and the domain, it is four separate questions.

In particular, I am interested in the semi-group of endomorphisms of $R(SL_2({\mathbb C}))$. I can see 3 of its elements: zero, identity and $$P: R(SL_2({\mathbb C}))\cong R(PSL_2({\mathbb C}))\hookrightarrow R(SL_2({\mathbb C})).$$ If one thinks of $x$ as the standard 2-dimensional representation of $SL_2$, they are given by $$0: x\mapsto 2, \ \ 1:x\mapsto x, \ \ P:x\mapsto x^2-2.$$ Question 2: Is this semigroup cyclic? Note that $P$ will be the generator.