The irreducibility of ladder determinantal varieties (Q1083500)

Let \({\mathfrak R}=(X_{ij})_{1\leq i\leq \mu,1\leq j\leq \nu}\) be a matrix of indeterminates over a domain k, where \(\mu,\nu\in {\mathbb{N}}\). Let \(p\in {\mathbb{N}}\) with \(1\leq p\leq \min (\mu,\nu)\). A subset \({\mathfrak L}\) of \({\mathfrak R}\) is called a ladder if whenever \(X_{ij},X_{k\ell}\in {\mathfrak L}\), \(i<k\), \(j<\ell\), we have \(X_{i\ell},X_{kj}\in {\mathfrak L}\). The author shows by combinatorial methods that the ideal \(I_ p({\mathfrak L})\subset k[{\mathfrak L}]\), generated by the size p minors of \({\mathfrak L}\), is prime (theorem 4.2). The basic fact for this result is theorem 2.45 which establishes that the size p minors of \({\mathfrak R}\) form a standard basis of \(I_ p({\mathfrak R})\); hence (s. corollary 3.4) the size p minors of \({\mathfrak L}\) form a standard basis of \(I_ p({\mathfrak L})\), too.