An insertion algorithm for catabolizability

Our recent work in [1] exhibits a canonical basis of the Garsia-Procesi module Rλ with cells labeled by standard tableaux of catabolizability D λ. Through our study of the Kazhdan-Lusztig preorder on this basis, we found a way to transform a standard word labeling a basis element into a word inserting to the unique tableau of shape λ. This led to an algorithm that computes the catabolizability of the insertion tableau of a standard word. We deduce from this a characterization of catabolizability as the statistic on words invariant under Knuth transformations, certain (co)rotations, and a new set of transformations we call catabolism transformations. We further deduce a Greene’s Theorem-like characterization of catabolizability and a result about how cocyclage changes catabolizability, strengthening a similar result in [8].