This article is cited in 1 paper

Lagrangian Grassmannians and Spinor Varieties in Characteristic Two

Bert van Geemen^a, Alessio Marrani<sup>bcd

a</sup> 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 INFN, sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy
^d Dipartimento di Fisica "Galileo Galilei", Università degli studi di Padova, I-35131 Padova, Italy

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.

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

MSC: 14M17, 20G15, 51E25

Received: March 8, 2019; in final form August 21, 2019; Published online August 27, 2019

Language: English

DOI: 10.3842/SIGMA.2019.064

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ArXiv: 1903.01228