Non-linearizable Root Group Data

Abstract

An RGD system $\mathcal{D}$ is called \emph{linear w.r.t. a root basis $\mathcal{B}$} if the commutation relations between the root groups of $\mathcal{D}$ are `linear' in a certain sense. Moreover, $\mathcal{D}$ is called \emph{linearizable}, if there exists a root basis $\mathcal{B}$ such that $\mathcal{D}$ is linear w.r.t. $\mathcal{B}$. For many examples of RGD systems it is easy to see that they are linear w.r.t. a concrete root basis. To the best of our knowledge, it was unclear whether RGD systems exist which are not linearizable. In this article we show that there exist uncountably many RGD systems which are not linearizable. In particular, we provide the first explicit example of such an RGD system. This expands the quote from R\'{e}my that axiom (RGD$1$)$_{\mathrm{lin}}$ is not only a strengthening of axiom (RGD$1$), but is in fact stronger than it. We show that non-linearizability appears in examples of universal type, and also in examples of $2$-spherical type. For the examples of universal type we construct an uncountable family of non-linearizable RGD systems, and for the examples of $2$-spherical type we show that the RGD systems of type $(4, 4, 4)$ recently constructed by the author provide uncountably many non-linearizable RGD systems.