The geometries of Jordan nets and Jordan webs

Abstract

A Jordan net (resp. web) is an embedding of a unital Jordan algebra of dimension 3 (resp. 4) into the space $${\mathbb{S}}^n$$ S n of symmetric $$n\times n$$ n × n matrices. We study the geometries of Jordan nets and webs: we classify the congruence orbits of Jordan nets (resp. webs) in $${\mathbb{S}}^n$$ S n for $$n\le 7$$ n ≤ 7 (resp. $$n\le 5$$ n ≤ 5 ), we find degenerations between these orbits and list obstructions to the existence of such degenerations. For Jordan nets in $$\mathbb{S}^n$$ S n for $$n\le 5$$ n ≤ 5 , these obstructions show that our list of degenerations is complete . For $$n=6$$ n = 6 , the existence of one degeneration is still undetermined. To explore further, we used an algorithm that indicates numerically whether a degeneration between two orbits exists. We verified this algorithm using all known degenerations and obstructions and then used it to compute the degenerations between Jordan nets in $$\mathbb {S}^7$$ S 7 and Jordan webs in $$\mathbb {S}^n$$ S n for $$n=4,5$$ n = 4 , 5 .