Determining the complexity of Kazhdan-Lusztig varieties

Kazhdan-Lusztig varieties are defined by ideals generated by certain minors of a matrix, which are chosen by a combinatorial rule. These varieties are of interest in commutative algebra and Schubert varieties. Each Kazhdan-Lusztig variety has a natural torus action from which one can construct a polytope. The complexity of this torus action can be computed from the dimension of the polytope and, in some sense, indicates how close the geometry of the variety is to the combinatorics of the associated polytope. In joint work with Maria Donten-Bury and Irem Portakal we address the problem of classifying which Kazhdan-Lusztig varieties have a given complexity. We do so by utilizing the rich combinatorics of Kazhdan-Lusztig varieties, which is reflected on the polytopes.