Logic and invariant theory. I. Invariant theory of projective properties
Walter J. Whiteley · Transactions of the American Mathematical Society · 1973
This paper initiates a series of papers which will reexamine some problems and results of classical invariant theory, within the framework of modern first-order logic. In this paper the notion that an equation is of invariant significance for the general linear group is extended in two directions. It is extended to define invariance of an arbitrary first-order formula for a category of linear transformations between vector spaces of dimension n. These invariant formulas are characterized by equivalence to formulas of a particular syntactic form: homogeneous formulas in determinants or “brackets". The fuller category of all semilinear transformations is also introduced in order to cover all changes of coordinates in a projective space. Invariance for this category is investigated. The results are extended to cover invariant formulas with both covariant and contravariant vectors. Finally, Klein’s Erlanger Program is reexamined in the light of the extended notion of invariance as well as some possible geometric categories.