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Automorphisms and opposition in spherical buildings of exceptional type, I

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Abstract
To each automorphism of a spherical building, there is a naturally associated opposition diagram, which encodes the types of the simplices of the building that are mapped onto opposite simplices. If no chamber (that is, no maximal simplex) of the building is mapped onto an opposite chamber, then the automorphism is called domestic. In this paper, we give the complete classification of domestic automorphisms of split spherical buildings of types E-6, F-4, and G(2). Moreover, for all split spherical buildings of exceptional type, we classify (i) the domestic homologies, (ii) the opposition diagrams arising from elements of the standard unipotent subgroup of the Chevalley group, and (iii) the automorphisms with opposition diagrams with at most two distinguished orbits encircled. Our results provide unexpected characterizations of long root elations and products of perpendicular long root elations in long root geometries, and analogues of the density theorem for connected linear algebraic groups in the setting of Chevalley groups over arbitrary fields.
Keywords
Exceptional spherical buildings, opposition diagram, domestic, automorphism

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MLA
Parkinson, James, and Hendrik Van Maldeghem. “Automorphisms and Opposition in Spherical Buildings of Exceptional Type, I.” CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, vol. 74, no. 6, 2022, pp. 1517–78, doi:10.4153/S0008414X21000341.
APA
Parkinson, J., & Van Maldeghem, H. (2022). Automorphisms and opposition in spherical buildings of exceptional type, I. CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 74(6), 1517–1578. https://doi.org/10.4153/S0008414X21000341
Chicago author-date
Parkinson, James, and Hendrik Van Maldeghem. 2022. “Automorphisms and Opposition in Spherical Buildings of Exceptional Type, I.” CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES 74 (6): 1517–78. https://doi.org/10.4153/S0008414X21000341.
Chicago author-date (all authors)
Parkinson, James, and Hendrik Van Maldeghem. 2022. “Automorphisms and Opposition in Spherical Buildings of Exceptional Type, I.” CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES 74 (6): 1517–1578. doi:10.4153/S0008414X21000341.
Vancouver
1.
Parkinson J, Van Maldeghem H. Automorphisms and opposition in spherical buildings of exceptional type, I. CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES. 2022;74(6):1517–78.
IEEE
[1]
J. Parkinson and H. Van Maldeghem, “Automorphisms and opposition in spherical buildings of exceptional type, I,” CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, vol. 74, no. 6, pp. 1517–1578, 2022.
@article{8755219,
  abstract     = {{To each automorphism of a spherical building, there is a naturally associated opposition diagram, which encodes the types of the simplices of the building that are mapped onto opposite simplices. If no chamber (that is, no maximal simplex) of the building is mapped onto an opposite chamber, then the automorphism is called domestic. In this paper, we give the complete classification of domestic automorphisms of split spherical buildings of types E-6, F-4, and G(2). Moreover, for all split spherical buildings of exceptional type, we classify (i) the domestic homologies, (ii) the opposition diagrams arising from elements of the standard unipotent subgroup of the Chevalley group, and (iii) the automorphisms with opposition diagrams with at most two distinguished orbits encircled. Our results provide unexpected characterizations of long root elations and products of perpendicular long root elations in long root geometries, and analogues of the density theorem for connected linear algebraic groups in the setting of Chevalley groups over arbitrary fields.}},
  author       = {{Parkinson, James and Van Maldeghem, Hendrik}},
  issn         = {{0008-414X}},
  journal      = {{CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES}},
  keywords     = {{Exceptional spherical buildings,opposition diagram,domestic,automorphism}},
  language     = {{eng}},
  number       = {{6}},
  pages        = {{1517--1578}},
  title        = {{Automorphisms and opposition in spherical buildings of exceptional type, I}},
  url          = {{http://doi.org/10.4153/S0008414X21000341}},
  volume       = {{74}},
  year         = {{2022}},
}

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