On Ideals of Minors of Matrices with Indeterminate Entries

This paper has two aims. The first is to study ideals of minors of matrices whose entries are among the variables of a polynomial ring. Specifically, we describe matrices whose ideals of minors of a given size are prime. The “generic” case, where all the entries are distinct variables has been studied extensively (cf. [1] and [2] for a thorough account.) While some special cases, such as catalecticant matrices and other 1-generic matrices, have been studied by other authors (e.g., [4]), the general case is not well understood. The main result in the first part of this paper is Theorem 2.3 which gives sufficient conditions for the ideal of minors of a matrix to be prime. This theorem is general enough to include interesting examples, such as the ideal of maximal minors of catalecticant matrices and their generalisations discussed in the second part of the paper. The second aim of this paper is to settle a specific problem raised by David Eisenbud and Frank-Olaf Schreyer (cf. [5]) on the primary decomposition of an ideal of maximal minors. We solve this problem by applying 2.3 together with some ad-hoc techniques. Throughout this paper K shall denote a field. For any matrix M with entries in a ring and any t ≥ 1, It(M) will denote the ideal generated by the t×t minors ofM . The results to be presented here rely on well known properties of determinantal rings which we summarise below: