Free resolutions from opposite Schubert varieties in minuscule homogeneous spaces

Free resolutions $F_\bullet$ of Cohen-Macaulay and Gorenstein ideals have been investigated for a long time. An important task is to determine generic resolutions for a given format ${rk F_i}$. Starting from the Kac-Moody Lie algebra associated to a T-shaped graph T_{p,q,r}, Weyman constructed generic rings for every format of resolutions of length 3. When the graph $T_{p,q,r}$ is Dynkin, these generic rings are Noetherian. Sam and Weyman showed that for all Dynkin types the ideals of the intersections of certain Schubert varieties of codimension 3 with the opposite big cell of the homogeneous spaces $G(T_{p,q,r})/P$, where $P$ is a specified maximal parabolic subgroup, have resolutions of the given format. In joint work with J. Torres and J. Weyman we study the case of Schubert varieties in minuscule homogeneous spaces and find resolutions of some well-known Cohen-Macaulay and Gorenstein ideals of higher codimension.