Ideals of adjacent minors (Q1882882)

Let \(X_{mn}\) be an \(m\times n\) matrix of indeterminates over a field \(K\). An adjacent \(k\times k\) minor is a minor of \(X_{mn}\) formed by \(k\) consecutive rows and columns. In the paper under review, the authors study the ideals \(I_{mn}(k)\) of \(K[x_{ij}]\) generated by all \(k\times k\) adjacent minors. They give a description of the minimal primes of the ideal generated by \(2\times 2\) adjacent minors. Also, they compute the complete prime decomposition of \(I_{mn}(m)\) when the characteristic of the ground field is 0. It turns out that \(I_{mn}(m)\) is a complete intersection and radical. As a result the paper gives a large new class of mixed determinantal ideals which are prime. The ideal \(I_{mn}(2)\) naturally appears in algebraic statistics. This motivates the authors to introduce also adjacent minors of a generic multidimensional matrix and to make the first steps in studying them.