Arguesian identities in invariant theory (Q1924005)

The paper begins with a long introduction including the following observation. ``The key of classical invariant theory \dots begins with the evident remark that the facts of geometry, when expressed in terms of algebra, become dependent upon the choice of a coordinate system. Yet a statement about space can be viewed as a geometric fact only on the condition that such a fact express a property of space which is independent of the choice of a coordinate system. \dots The emphasis of classical invariant theory lies not in the description of geometric objects, such as algebraic varieties by coordinate-free devices, but in the discovery of a variety of new operations that can be defined among geometric entities in a coordinate-free manner.'' In this spirit, the remainder of the paper presents a technique for construction of geometric identities in a Grassmann-Cayley algebra. Each of these identities represents a projective invariant closely related to an \(n\)-dimensional Desargues Theorem and is proved by purely algebraic manipulation. The main theorem (Theorem 4.1) links the construction of such identities with calculations in a bipartite graph. The work was motivated in part by a paper of \textit{P. Doubilet}, \textit{G.-C. Rota} and \textit{J. Stein} [Stud. Appl. Math. 53, 185-216 (1974; Zbl 0426.05009)].