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A geometric connection between the split first and second rows of the Freudenthal-Tits magic square

Anneleen De Schepper and Magali Victoor
(2023) INNOVATIONS IN INCIDENCE GEOMETRY - ALGEBRAIC, TOPOLOGICAL AND COMBINATORIAL. 20(1). p.1-53
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Abstract
A projective representation $G_1$ of a variety of the first row of the Freudenthal–Tits magic square can be obtained as the absolute geometry of a (symplectic) polarity $\rho$ of the projective representation $G_2$ of a variety one cell below. In this paper, we extend this geometric connection between $G_1$ and $G_2$ by showing that any nondegenerate quadric $Q$ of maximal Witt index containing $G_2$ gives rise to a variety isomorphic to $G_1$, in the sense that the symplecta of $G_2$ contained in totally isotropic subspaces of $Q$ are the absolute symplecta of a unique (symplectic) polarity $\rho$ of $G_2$. Except for the smallest case, we also show that any nondegenerate quadric containing $G_2$ has maximal Witt index; and in the largest case, we obtain that there are only three kinds of possibly degenerate quadrics containing the Cartan variety $\mathcal{E}_6(\mathbb{K})$.
Keywords
Veronese variety, Spherical buildings, Embeddings, Geometric hyperplanes

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MLA
De Schepper, Anneleen, and Magali Victoor. “A Geometric Connection between the Split First and Second Rows of the Freudenthal-Tits Magic Square.” INNOVATIONS IN INCIDENCE GEOMETRY - ALGEBRAIC, TOPOLOGICAL AND COMBINATORIAL, vol. 20, no. 1, 2023, pp. 1–53, doi:10.2140/iig.2023.20.1.
APA
De Schepper, A., & Victoor, M. (2023). A geometric connection between the split first and second rows of the Freudenthal-Tits magic square. INNOVATIONS IN INCIDENCE GEOMETRY - ALGEBRAIC, TOPOLOGICAL AND COMBINATORIAL, 20(1), 1–53. https://doi.org/10.2140/iig.2023.20.1
Chicago author-date
De Schepper, Anneleen, and Magali Victoor. 2023. “A Geometric Connection between the Split First and Second Rows of the Freudenthal-Tits Magic Square.” INNOVATIONS IN INCIDENCE GEOMETRY - ALGEBRAIC, TOPOLOGICAL AND COMBINATORIAL 20 (1): 1–53. https://doi.org/10.2140/iig.2023.20.1.
Chicago author-date (all authors)
De Schepper, Anneleen, and Magali Victoor. 2023. “A Geometric Connection between the Split First and Second Rows of the Freudenthal-Tits Magic Square.” INNOVATIONS IN INCIDENCE GEOMETRY - ALGEBRAIC, TOPOLOGICAL AND COMBINATORIAL 20 (1): 1–53. doi:10.2140/iig.2023.20.1.
Vancouver
1.
De Schepper A, Victoor M. A geometric connection between the split first and second rows of the Freudenthal-Tits magic square. INNOVATIONS IN INCIDENCE GEOMETRY - ALGEBRAIC, TOPOLOGICAL AND COMBINATORIAL. 2023;20(1):1–53.
IEEE
[1]
A. De Schepper and M. Victoor, “A geometric connection between the split first and second rows of the Freudenthal-Tits magic square,” INNOVATIONS IN INCIDENCE GEOMETRY - ALGEBRAIC, TOPOLOGICAL AND COMBINATORIAL, vol. 20, no. 1, pp. 1–53, 2023.
@article{01H118M3X94CMSGPTX5Z4A20KV,
  abstract     = {{A projective representation $G_1$ of a variety of the first row of the Freudenthal–Tits magic square can be obtained as the absolute geometry of a (symplectic) polarity $\rho$ of the projective representation $G_2$ of a variety one cell below. In this paper, we extend this geometric connection between $G_1$ and $G_2$ by showing that any nondegenerate quadric $Q$ of maximal Witt index containing $G_2$ gives rise to a variety isomorphic to $G_1$, in the sense that the symplecta of $G_2$ contained in totally isotropic subspaces of $Q$ are the absolute symplecta of a unique (symplectic) polarity $\rho$ of $G_2$. Except for the smallest case, we also show that any nondegenerate quadric containing $G_2$ has maximal Witt index; and in the largest case, we obtain that there are only three kinds of possibly degenerate quadrics containing the Cartan variety $\mathcal{E}_6(\mathbb{K})$.}},
  author       = {{De Schepper, Anneleen and Victoor, Magali}},
  issn         = {{2640-7337}},
  journal      = {{INNOVATIONS IN INCIDENCE GEOMETRY - ALGEBRAIC, TOPOLOGICAL AND COMBINATORIAL}},
  keywords     = {{Veronese variety,Spherical buildings,Embeddings,Geometric hyperplanes}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{1--53}},
  title        = {{A geometric connection between the split first and second rows of the Freudenthal-Tits magic square}},
  url          = {{http://doi.org/10.2140/iig.2023.20.1}},
  volume       = {{20}},
  year         = {{2023}},
}
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