Arguesian Identities in Linear Lattices

✍ Scribed by Matteo Mainetti; Catherine Huafei Yan

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 486 KB
• Volume: 144
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1998.1818

✦ Synopsis

A class of identities in the Grassmann Cayley algebra which yields a large number of geometric theorems on the incidence of subspaces of projective spaces was found by Hawrylycz (``Geometric Identities in Invariant Theory,'' Ph.D. thesis, Massachusetts, Institute of Technology, 1994). In this paper we establish a link between such identities in the Grassmann Cayley algebra and a class of inequalities in the class of linear lattices, i.e., the lattices of commuting equivalence relations. We prove that a subclass of identities found by Hawrylycz, namely, the Arguesian identities of order 2, can be systematically translated into inequalities holding in linear lattices. As a consequence, we obtain a family of geometric theorems on the incidence of subspaces that are characteristic-free and independent of dimensions.