Minors of Symmetric and Exterior Powers

We describe some of the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski's theory of complete ideals and of representation theory. Let R be a commutative ring. The determinantal ideals attached to matrices with entries in R play ubiquitous roles in the study of the syzygies of R{modules. In this note, we describe some of the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski's theory of complete ideals and of representation theory, the results being sharper for rings containing the rationals. Let R be an integral domain (or a eld) and ' : R ! R anR-linear map of rank r. It is an easy exercise to show that the d-th symmetric power S(') : S(R) ! S(R) has rank r+d 1 d . Let It(') denote the ideal generated by the minors (of a matrix representing '). Since rank' = maxfr : Ir(') 6= 0g; one can immediately determine the radicals of the ideals It(S (')), namely Rad It(S (')) = Rad Ir(') if r + d 1 d t < r + d