Automorphisms of \(\det (X_{ij}):\) The group scheme approach
Let A be a commutative ring, let \(X=(X_{ij})_{1\leq i,j\leq n}\) be a matrix of indeterminates over A, and let det(X) denote its determinant in the polynomial ring A[X]. The problem treated in this paper is to determine the subgroup of GL(n 2,A), consisting of those linear transformations of the variables, which preserve det(X). As the author notes, the result over fields is due to Frobenius, has been rediscovered in that context several times, and has been obtained in general by \textit{B. R. McDonald} [Mem. Am. Math. Soc. 287 (1983; Zbl 0529.15011)]. However, as underlined in the title, the approach used here is based on group scheme techniques. In fact, it is forcefully stated in the introduction: ``This is also a programmatic paper, and the results on determinants are only part of this raison d'être'', and further: ``...I want to make it easy for others to use the methods of this paper in their own work. To this end I have concentrated all the general group scheme arguments in the first section, where they are woven into a brief exposition of the properties of affine group schemes.'' At least as far as this reviewer is concerned, the evidence offered by the article for the power and flexibility of the chosen approach is very conclusive: besides solving the original problem, it does the same for the corresponding one with X a generic symmetric matrix, and determines the automorphisms preserving the Pfaffian of a generic alternating matrix. It also contains a description of the groups of automorphisms of the ideals generated by the minors (or Pfaffians) of X of a fixed order.
scientific article; zbMATH DE number 4061402
scientific article