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MiS Preprint
1/2022

The Geometries of Jordan nets and Jordan webs

Arthur Bik and Henrik Eisenmann

Abstract

A Jordan net (resp. web) is an embedding of a unital Jordan algebra of dimension $3$ (resp. $4$) into the space $\mathbb S^n$ of symmetric $n\times n$ matrices. We study the geometries of Jordan nets and webs: we classify the congruence-orbits of Jordan nets (resp. webs) in $\mathbb S^n$ for $n\leq 7$ (resp. $n\leq 5$), we find degenerations between these orbits and list obstructions to the existence of such degenerations. For Jordan nets in $\mathbb S^n$ for $n\leq5$, these obstructions show that our list of degenerations is complete. For $n=6$, 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 $\mathbb S^7$ and Jordan webs in $\mathbb S^n$ for $n=4,5$.

Received:
12.01.2022
Published:
12.01.2022
MSC Codes:
17C50, 14M15, 14L30, 65K10

Related publications

inJournal
2022 Journal Open Access
Arthur Bik and Henrik Eisenmann

The geometries of Jordan nets and Jordan webs

In: Annali di matematica pura ed applicata, 201 (2022) 5, pp. 2413-2464