Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 064, 22 pages      arXiv:1903.01228      https://doi.org/10.3842/SIGMA.2019.064

Lagrangian Grassmannians and Spinor Varieties in Characteristic Two

Bert van Geemen a and Alessio Marrani bc
a) Dipartimento di Matematica, Università di Milano, Via Saldini 50, I-20133 Milano, Italy
b) Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi, Via Panisperna 89A, I-00184, Roma, Italy
c) Dipartimento di Fisica e Astronomia Galileo Galilei, Università di Padova, and INFN, sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy

Received March 08, 2019, in final form August 21, 2019; Published online August 27, 2019

Abstract
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.

Key words: Lagrangian Grassmannian; spinor variety; characteristic two; Freudenthal triple system.

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