Properties of a 27-dimensional space of symmetric bilinear forms acted on by E6

Abstract

We use the 27-dimensional simple exceptional Jordan algebra to construct a 27-dimensional space of symmetric bilinear forms invariant under the action of the simple Lie group E 6. Using the E 6 action on the Jordan algebra, we show that the space contains elements of exactly three non-zero ranks, namely, 10, 18 and 27. Many of the properties of the Jordan algebra are reflected in rank properties of the space of forms. We investigate subspaces containing no elements of rank 27 and also constant rank subspaces. The results vary depending on the nature of the octonion algebra used to define the Jordan algebra. © Springer Science+Business Media, LLC 2012.