Affine Grassmannians for Triality Groups

TL;DR

This paper explores affine Grassmannians associated with ramified triality groups of type ${}^3D_4$, describing their structure via automorphisms of twisted composition algebras derived from octonions.

Contribution

It provides a new description of affine Grassmannians for triality groups using twisted composition algebras and lattice classifications.

Findings

Affine Grassmannians characterized as functors classifying lattices.

Connection established between triality groups and octonion-based composition algebras.

New geometric descriptions for ramified triality groups.

Abstract

We study affine Grassmannians for ramified triality groups. These groups are of type ${}^3D_4$, so they are forms of the orthogonal or the spin groups in 8 variables. They can be given as automorphisms of certain twisted composition algebras obtained from the octonion algebra. Using these composition algebras, we give descriptions of the affine Grassmannians for these triality groups as functors classifying suitable lattices in a fixed space.