Lagrangian Grassmannians and Spinor Varieties in Characteristic Two

Symmetry Integrability and Geometry-Methods and Applications

article

Bert van Geemen¹  Alessio Marrani² 

[1] Dipartimento di Matematica, Università di Milano;Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi

关键词: Lagrangian Grassmannian;    spinor variety;    characteristic two;    Freudenthal triple system;   

DOI  :  10.3842/SIGMA.2019.064

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown