# Lagrangian Grassmannians and Spinor Varieties in Characteristic Two

## TL;DR

This paper investigates the equations defining the Lagrangian Grassmannian's image under principal minors in characteristic two, revealing it is characterized by quadrics and is isomorphic to a spinor variety for all n.

## Contribution

It proves that in characteristic two, the image of the Lagrangian Grassmannian under principal minors is defined by quadrics for all n and identifies it with the spinor variety associated to Spin(2n+1).

## Key findings

- The image is defined by quadrics in characteristic two.
- The image is isomorphic to the spinor variety for all n.
- Connections to supergravity and black-hole/qubit correspondence are discussed.

## 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.