Resolutions of determinantal ideals: The submaximal minors

Abstract

For many years there has been considerable interest in finding a resolution of the ideal generated by the minors of order p of a generic m x n matrix. To put the problem more precisely, suppose R, is a commutative ring and X, are variables with 1 < i < m and 1 < j Q n. If we let R = R, [X,] be the polynomial ring over R,, then we have the “generic” matrix (X,) and we may form the ideal ZP in R generated by the p X p minors of this matrix. The problem, then, is to find an explicit free resolution of the ideal ZP over the ring R. It was proved by Eagon and Hochster [lo] that R/I, has a resolution of length (m-p + l)(n -p + l), but their proof consisted in showing that the ideal ZP is perfect; it did not provide a construction of the resolution. In fact, it is not known whether the Betti numbers of the ideal ZP depend on the characteristic of the ground ring R,. In [ 121, Lascoux succeeded in giving an explicit resolution provided that the ground ring R, contained the field, Q, of rational numbers. His construction rests heavily on the theory of Schur functors and the fact that in characteristic zero the Schur functors are the irreducible representations of the general linear group. Over the integers, however, the construction breaks down despite the fact that one can define the Schur functors over an arbitrary commutative ring (see [ 1,2, 14, 151 for various constructions of Schur functors). In analyzing the work of Lascoux and its subsequent reworking by Nielsen [ 131, some basic facts seemed to clamor for attention. One was that within a resolution of R/Z,, there appeared to be two types of boundary maps: one of degree 1 and one of degree p. The maps of degree 1 were maps between sums of Schur functors of fixed Durfee square k (see Section 2 for definitions), while the maps of degree p were from sums of Schur functors of