Arguesian Identities in Invariant Theory

Abstract

Having been motivated by an example of Doubilet, Rota, and Stein [Stud. Appl. Math.56(1976), 185–216], we present a technique for constructing geometric identities in a Grassmann–Cayley algebra. Each identity represents a projective invariant closely related to the Theorem of Desargues in the plane and its generalizations to higher dimensional projective space. The construction employs certain combinatorial properties of matchings in bipartite graphs. We also prove a dimension independence result for Arguesian identities, thereby connecting the identities with lattice theory.