THE EXCEPTIONAL TITS QUADRANGLES
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Springer Science+Business Media New York (2020)
THE EXCEPTIONAL TITS QUADRANGLES ∗ ¨ BERNHARD MUHLHERR
RICHARD M. WEISS∗∗
Mathematisches Institut Universit¨at Giessen 35392 Giessen, Germany
Department of Mathematics Tufts University Medford, MA 02155, USA
[email protected]
[email protected]
Abstract. A Tits polygon is a bipartite graph in which the neighborhood of each vertex is endowed with an “opposition relation” satisfying certain axioms. Moufang polygons are precisely the Tits polygons in which these opposition relations are all trivial. Every Tits polygon has a distinguished set of circuits. A Tits quadrangle is a Tits polygon in which these circuits all have length 8. There is a standard construction that produces a Tits polygon from certain pairs (∆, T ), where ∆ is an irreducible spherical building and T is a Tits index of relative rank 2. We call a Tits quadrangle exceptional if it arises from such a pair (∆, T ) for ∆ the spherical building associated to the group of rational points of an exceptional algebraic group. In this paper, we characterize the exceptional Tits quadrangles as extensions of orthogonal Tits quadrangles in a suitable sense.
1. Introduction A generalized polygon is the same thing as a spherical building of rank 2. Generalized polygons are too numerous to classify, but Tits observed that the generalized polygons associated with absolutely simple algebraic groups of Krank 2 satisfy a symmetry property he called the Moufang condition. Moufang polygons (that is to say, generalized polygons satisfying the Moufang condition) were classified in [10]. In [7], we introduced the notion of a Tits polygon. A Tits polygon is a bipartite graph in which the neighborhood of each vertex is endowed with an “opposition relation” satisfying certain axioms. Moufang polygons are precisely the Tits polygons in which these opposition relations are all trivial. Every Tits polygon has a distinguished set of circuits of some fixed length. A Tits n-gon is a Tits polygon in which these circuits all have length 2n. Let I denote the set of pairs (∆, T ), where ∆ is an irreducible spherical building of rank m ≥ 2 (assumed to be a Moufang polygon if m = 2) and T is a Tits index of relative rank 2 whose absolute type is the Coxeter system of ∆. Every pair (∆, T ) in I gives rise, in a natural way, to a Tits polygon X∆,T . We call the Tits polygons DOI: 10.1007/S00031-020-09573-5 by a grant from the DFG Supported by a Collaboration Grant from the Simons Foundation. Received June 11, 2018. Accepted August 7, 2019. Corresponding Author: Richard M. Weiss, e-mail: [email protected] ∗ Supported ∗∗
¨ BERNHARD MUHLHERR, RICHARD M. WEISS
that arise in this way the Tits polygons of index type. For each (∆, T ) ∈ I, we have X∆,T = ∆ if ∆ is a Moufang polygon, but if the rank of ∆ is at least 3, the opposition relations of X∆,T are not all trivial. Almost all Tits polygons of index type satisfy a natural condition involving the normalizers of its root groups. We call a Tits polygon dagger-sharp i
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