Construction of Commutator Blueprints

Commutator blueprints can be seen as blueprints for constructing RGD-systems over $\mathbb{F}_2$ with prescribed commutation relations. In this paper we construct several families of Weyl-invariant commutator blueprints, mostly of universal type. Together with the main result of \cite{BiRGD} we obtain new examples of exotic RGD-systems of universal type over $\mathbb{F}_2$.

💡 Research Summary

The paper introduces the notion of a “commutator blueprint”, a purely combinatorial datum that records prescribed commutation relations among root groups in an RGD‑system. An RGD‑system is a group equipped with a family of subgroups indexed by the real roots of a Coxeter system, satisfying axioms modeled on Kac‑Moody groups. The central requirement for a blueprint to be realizable (“integrable”) is Weyl‑invariance: the prescribed relations must be stable under the action of the Weyl group.

To make the construction tractable the author first defines a weaker object, a “pre‑commutator blueprint”, which only needs to satisfy certain length and nilpotency conditions. Lemma 3.5 and Theorem 3.8 give sufficient criteria: if the groups (U_w) generated by the root groups have order (2^{\ell(w)}) (where (\ell) is the Coxeter length) and are nilpotent of class at most 2, then the pre‑blueprint already qualifies as a full commutator blueprint. This technical groundwork underlies all subsequent constructions.

Four main families of Weyl‑invariant commutator blueprints are produced.

Theorem A (Universal type). For any Coxeter system ((W,S)) of universal type (all distinct generators have infinite order product), the author selects a pair (s\neq t) and builds a blueprint whose restriction to the rank‑2 residue ({s,t}) reproduces the commutation relations of the trivalent Moufang twin trees originally constructed by Tits. This yields, via the main result of