Subspaces of 7 x 7 skew-symmetric matrices related to the group G₂

2008/11/08 by Gow, Roderick
#17A36 #FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.0811.1298

Abstract

Let K be a field of characteristic different from 2 and let C be an octonion algebra over K. We show that there is a seven-dimensional subspace of 7× 7 skew-symmetric matrices over K which is invariant under the automorphism group of C. This subspace consists of elements of rank 6 when C is a division algebra, and elements of rank 4 and 6 when C is a split algebra. In the latter case, the automorphism group is the exceptional group G₂(K).

Related

• A unital in the generalized hexagon of order two, and an exceptional isomorphism between finite groups of Lie type
• Lattice Filtrations for G₂ of a p-adic Field
• Isomorphism classes of k-involutions of algebraic groups of type E₆
• SL(2) over the octonions
• Invariants of G₂ and Spin(7) in positive characteristic