Lagrangian Grassmannians and Spinor Varieties in Characteristic Two

Bert van Geemen   ; Alessio Marrani

Symmetry, integrability and geometry: methods and applications, Tome 15 (2019)

Résumé

The vector space of symmetric matrices of size $n$ has a natural map to a projective space of dimension $2^n-1$ given by the principal minors. This map extends to the Lagrangian Grassmannian ${\rm LG}(n,2n)$ and over the complex numbers the image is defined, as a set, by quartic equations. In case the characteristic of the field is two, it was observed that, for $n=3,4$, the image is defined by quadrics. In this paper we show that this is the case for any $n$ and that moreover the image is the spinor variety associated to ${\rm Spin}(2n+1)$. Since some of the motivating examples are of interest in supergravity and in the black-hole/qubit correspondence, we conclude with a brief examination of other cases related to integral Freudenthal triple systems over integral cubic Jordan algebras.

Détail

Keywords: Lagrangian Grassmannian, spinor variety, characteristic two, Freudenthal triple system.

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