The Cayley Grassmannian

TL;DR

The paper investigates the Cayley Grassmannian, a complex algebraic variety parametrizing four-dimensional subalgebras of octonions, revealing its structure as a spherical G2-variety with explicit orbit and cohomology descriptions.

Contribution

It establishes the Cayley Grassmannian as a spherical G2-variety with three orbits and fully determines its cohomology ring and intersection product.

Findings

Cayley Grassmannian is a spherical G2-variety.

It has exactly three orbits.

The cohomology ring and intersection product are explicitly computed.

Abstract

We study the projective variety CG parametrizing four dimensional subalgebras of the complex octonions, which we call the Cayley Grass-mannian. We prove that it is a spherical G2-variety with only three orbits that we describe explicitely. Its cohomology ring has a basis of Schubert type classes and we determine the intersection product completely.