Gröbner bases of powers of ideals of maximal minors
Let \(K\) be a field and let \(I=I_m\subset K[{\mathbf X}]\) denote the ideal of the polynomial ring \(K[{\mathbf X}]\) generated by the maximal minors of the \(m\times n\) (\(m\leq n\)) generic matrix \(({\mathbf X})\). It is a well known repeatedly proved result that the maximal minors themselves form a Gröbner basis of \(I\) under the so called diagonal order of the monomials of \(K[{\mathbf X}]\). The present paper extends this result to the powers of \(I\), namely, a Gröbner basis of the power \(I^r\) is constituted by the \(r\)-products of maximal minors. The main tool is the Knuth-Robinson-Schensted correspondence. A formula for the Hilbert series of the residue rings \(K[{\mathbf X}]/I^r\) is also given.
scientific article; zbMATH DE number 1089291