Tits pentagons and the root system \(GH_2\) (Q6654983)

A generalized polygon is a bipartite graph whose girth is twice its diameter, such objects are also known as spherical building of rank 2 (compare to Definition 4.20 in [\textit{P. Abramenko} and \textit{K. S. Brown}, Buildings. Theory and applications. Berlin: Springer (2008; Zbl 1214.20033)]). \textit{J. Tits}, in [Moufang polygons. Berlin-Heidelberg-New York: Springer-Verlag (1974; Zbl 0295.20047)], observed that the generalized polygons associated with absolutely simple algebraic groups, as well as all irreducible buildings of rank greater than \(2\), all enjoy a symmetry property, called the Moufang condition. Furthermore Tits proposed two projects in these addenda: to classify all Moufang polygons (i.e. all Moufang buildings of rank \(2\)), and then to use this to give a simpler and more uniform proof for the classification of spherical buildings of rank at least \(3\). Both projects are executed in the book of \textit{J. Tits} and the second author [Moufang polygons. Berlin: Springer (2002; Zbl 1010.20017)].\N\NA Tits polygon is a bipartite graph \(\Gamma\) in which for each vertex \(v\), the set \(\Gamma_{v}\) of vertices adjacent to \(v\) is endowed with an opposition relation satisfying certain axioms that reduce to the axioms of a Moufang polygon when these opposition relations are all simply the relation \( \not =\) on \(\Gamma_{v}\). The authors, with \textit{H. P. Petersson}, in [Tits Polygons. Providence, RI: AMS (2022; Zbl 1498.51001)], proved that sharp Tits \(n\)-gons exist only for \(n \in \{ 3, 4, 6, 8\}\).\N\NIn the paper under review, the authors turn to the first non-crystallographic value of \(n\). There is a generalization of the construction of a Tits polygon of index type that requires, in place of a Tits index, only an admissible partition as defined by the first author in [Lond. Math. Soc. Lect. Note Ser. 191, 277--287 (1993; Zbl 0820.20048)]. This construction produces Tits \(n\)-gons for every \(n\). In particular, it produces a Tits pentagon in the case that \(\Delta\) is a building of type \(\mathsf{A}_{4}\) and the admissible partition is \(\bullet\!\!-\!\!\circ\!\!-\!\!\bullet\!\!-\!\!\circ\). The building \(\Delta\) can be replaced by the geometry of a free module of rank 5 over a unitary associative ring of stable range 1 to produce an even larger family of Tits pentagons. In the paper under review the authors confirm that these Tits pentagons exist and prove that every 4-plump Tits pentagon is isomorphic to one of them.\N\NTo conclude the historical overview, the reviewer fully includes the last paragraph of the introduction.\N\N``\textit{Historical Note}: The last step in Tits' classification of irreducible spherical buildings of rank at least \(3\) was to show that there are no buildings whose Coxeter diagram is \(\mathsf{H}_{3}\) or \(\mathsf{H}_{4}\). This was the main theorem of [\textit{J. Tits}, Invent. Math. 43, 283--295 (1977; Zbl 0399.20037)], published a few years after the rest of the classification appeared in LNM 386 [Zbl 0295.20047]. Tits deduced the main theorem of [\textit{loc. cit.}] as a consequence of his result that there are no Moufang pentagons. He published a generalization of this result in [\textit{J. Tits}, Invent. Math. 36, 275--284 (1976; Zbl 0369.20004)]. It was this generalization that opened for Tits the possibility of classifying arbitrary Moufang polygons.''