Castelnuovo-Mumford regularity for $321$-avoiding Kazhdan-Lusztig varieties

Abstract: We introduce an algorithm which combinatorially computes the Castelnuovo-Mumford regularity of 321-avoiding Kazhdan-Lusztig varieties using excited diagrams. This extends earlier work of Rajchgot, Weigandt, and the author (2022) which computes the regularity of Grassmannian Kazhdan-Lusztig varieties. Our results specialize to compute the regularity of all two-sided mixed ladder determinantal varieties in terms of lattice paths.