Mixed Ladder Determinantal Varieties

We investigate ladder determinantal varieties defined by ideals of minors of possibly different sizes in the different regions (the steps) of one-sided ladders L. These varieties are an important generalization of the classical ladder determinantal varieties (i.e., with equal-size minors) since they are very closely related to Schubert varieties, this being the first main result of this paper. We show that they correspond to opposite cells in Schubert varieties in flag varieties of type An. In consequence, one deduces the normality and the Cohen–Macaulayness of these one-sided ladder determinantal varieties with ideals of mixed-size minors, as well as the fact that they have rational singularities. Next we show that, up to products by affine spaces, each of these varieties is a basic open set in a classical ladder determinantal variety (i.e., with equal-size minors) and that it contains as a basic open set another classical ladder determinantal variety. This result, along with a general localisation lemma used to show it, enables us to compute the divisor class group and singular locus of the coordinate rings of these varieties, as well as to determine when they are Gorenstein.