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Construction and characterisation of the varieties of the third row of the Freudenthal-Tits magic square

Anneleen De Schepper, Jeroen Schillewaert (UGent) , Hendrik Van Maldeghem (UGent) and Magali Victoor (UGent)
(2024) GEOMETRIAE DEDICATA. 218(1).
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Abstract
We characterise the varieties appearing in the third row of the Freudenthal-Tits magic square over an arbitrary field, in both the split and non-split version, as originally presented by Jacques Tits in his Habilitation thesis. In particular, we characterise the variety related to the 56-dimensional module of a Chevalley group of exceptional type E7 over an arbitrary field. We use an elementary axiom system which is the natural continuation of the one characterising the varieties of the second row of the magic square. We provide an explicit common construction of all characterised varieties as the quadratic Zariski closure of the image of a newly defined affine dual polar Veronese map. We also provide a construction of each of these varieties as the common null set of quadratic forms.
Keywords
REPRESENTATIONS, Lie incidence geometry, Dual polar spaces, Veronese variety, Buildings

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Citation

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MLA
De Schepper, Anneleen, et al. “Construction and Characterisation of the Varieties of the Third Row of the Freudenthal-Tits Magic Square.” GEOMETRIAE DEDICATA, vol. 218, no. 1, 2024, doi:10.1007/s10711-023-00864-1.
APA
De Schepper, A., Schillewaert, J., Van Maldeghem, H., & Victoor, M. (2024). Construction and characterisation of the varieties of the third row of the Freudenthal-Tits magic square. GEOMETRIAE DEDICATA, 218(1). https://doi.org/10.1007/s10711-023-00864-1
Chicago author-date
De Schepper, Anneleen, Jeroen Schillewaert, Hendrik Van Maldeghem, and Magali Victoor. 2024. “Construction and Characterisation of the Varieties of the Third Row of the Freudenthal-Tits Magic Square.” GEOMETRIAE DEDICATA 218 (1). https://doi.org/10.1007/s10711-023-00864-1.
Chicago author-date (all authors)
De Schepper, Anneleen, Jeroen Schillewaert, Hendrik Van Maldeghem, and Magali Victoor. 2024. “Construction and Characterisation of the Varieties of the Third Row of the Freudenthal-Tits Magic Square.” GEOMETRIAE DEDICATA 218 (1). doi:10.1007/s10711-023-00864-1.
Vancouver
1.
De Schepper A, Schillewaert J, Van Maldeghem H, Victoor M. Construction and characterisation of the varieties of the third row of the Freudenthal-Tits magic square. GEOMETRIAE DEDICATA. 2024;218(1).
IEEE
[1]
A. De Schepper, J. Schillewaert, H. Van Maldeghem, and M. Victoor, “Construction and characterisation of the varieties of the third row of the Freudenthal-Tits magic square,” GEOMETRIAE DEDICATA, vol. 218, no. 1, 2024.
@article{01HW5NM4RHW495SN5872W5C45T,
  abstract     = {{We characterise the varieties appearing in the third row of the Freudenthal-Tits magic square over an arbitrary field, in both the split and non-split version, as originally presented by Jacques Tits in his Habilitation thesis. In particular, we characterise the variety related to the 56-dimensional module of a Chevalley group of exceptional type E7 over an arbitrary field. We use an elementary axiom system which is the natural continuation of the one characterising the varieties of the second row of the magic square. We provide an explicit common construction of all characterised varieties as the quadratic Zariski closure of the image of a newly defined affine dual polar Veronese map. We also provide a construction of each of these varieties as the common null set of quadratic forms.}},
  articleno    = {{20}},
  author       = {{De Schepper, Anneleen and Schillewaert, Jeroen and Van Maldeghem, Hendrik and Victoor, Magali}},
  issn         = {{0046-5755}},
  journal      = {{GEOMETRIAE DEDICATA}},
  keywords     = {{REPRESENTATIONS,Lie incidence geometry,Dual polar spaces,Veronese variety,Buildings}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{57}},
  title        = {{Construction and characterisation of the varieties of the third row of the Freudenthal-Tits magic square}},
  url          = {{http://doi.org/10.1007/s10711-023-00864-1}},
  volume       = {{218}},
  year         = {{2024}},
}
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