Max Horn

Abstract involutions of algebraic groups and of Kac–Moody groups

Abstract Based on the second authors thesis [Horn, Involutions of Kac–Moody groups, TU Darmstadt, 2008], in this article we provide a uniform treatment of abstract involutions of algebraic groups and of Kac–Moody groups using twin buildings, RGD systems, and twisted involutions of Coxeter groups. Notably we simultaneously generalize the double coset decompositions established in [Helminck, Wang, Adv. Math. 99: 26–96, 1993] and [Springer, Algebraic groups and related topics: 525–543, North-Holland, 1984] for algebraic groups and in [Kac, Wang, Adv. Math. 92: 129–195, 1992] for certain Kac–Moody groups, we analyze the filtration studied in [Devillers, Mühlherr, Forum Math. 19: 955–970, 2007] in the context of arbitrary involutions, and we answer a structural question on the combinatorics of involutions of twin buildings raised in [Bennett, Gramlich, Hoffman, Shpectorov, Curtis–Phan–Tits theory: 13–29, World Scientific, 2003].

The complete Phan-type theorem for Sp(2n,q)

Abstract The articles [R. Gramlich, C. Homan and S. Shpectorov. A Phan-type theorem for Sp(2n; q). J. Algebra 264 (2003), 358–384.] and [R. Gramlich. Weak Phan systems of type C n . J. Algebra 280 (2004), 1–19.] give a characterization of central quotients of the group Sp(2n, q) for n ≥ 3 and all prime powers q, up to some small cases that are left open. The present article fills in this gap, thus providing the definitive version of the Phan-type theorem for Sp(2n, q).

Spin covers of maximal compact subgroups of Kac–Moody groups and spin-extended Weyl groups

Abstract Let G be a split real Kac–Moody group of arbitrary type and let K be its maximal compact subgroup, i.e. the subgroup of elements fixed by a Cartan–Chevalley involution of G. We construct non-trivial spin covers of K, thus confirming a conjecture by Damour and Hillmann. For irreducible simply-laced diagrams and for all spherical diagrams these spin covers are two-fold central extensions of K. For more complicated irreducible diagrams these spin covers are central extensions by a finite 2-group of possibly larger cardinality. Our construction is amalgam-theoretic and makes use of the generalised spin representations of maximal compact subalgebras of split real Kac–Moody algebras studied by Hainke, Levy and the third author. Our spin covers contain what we call spin-extended Weyl groups which admit a presentation by generators and relations obtained from the one for extended Weyl groups by relaxing the condition on the generators so that only their eighth powers are required to be trivial.

Intransitive geometries and fused amalgams

Abstract We study geometries that arise from the natural G2() action on the geometry of one-dimensional subspaces, of non-singular two-dimensional subspaces, and of non-singular three-dimensional subspaces of the building geometry of type C 3(), where is a perfect field of characteristic 2. One of these geometries is intransitive in such a way that the non-standard geometric covering theory from [R. Gramlich and H. Van Maldeghem. Intransitive geometries. Proc. London Math. Soc. (2) 93 (2006), 666–692.] is not applicable. In this paper we introduce the concept of fused amalgams in order to extend the geometric covering theory so that it applies to that geometry. This yields an interesting new amalgamation result for the group G2().