2023 journal article
RSK tableaux and the weak order on fully commutative permutations
ELECTRONIC JOURNAL OF COMBINATORICS, 30(4).
author keywords: boolean permutation; fully commutative permutation; permutation pattern; Robinson-Schensted-Knuth correspondence; reduced word; weak order
www.combinatorics.org (publisher PDF)
Source: Web Of Science
Added: December 18, 2023
For each fully commutative permutation, we construct a “boolean core,” which is the maximal boolean permutation in its principal order ideal under the right weak order. We partition the set of fully commutative permutations into the recently defined crowded and uncrowded elements, distinguished by whether or not their RSK insertion tableaux satisfy a sparsity condition. We show that a fully commutative element is uncrowded exactly when it shares the RSK insertion tableau with its boolean core. We present the dynamics of the right weak order on fully commutative permutations, with particular interest in when they change from uncrowded to crowded. In particular, we use consecutive permutation patterns and descents to characterize the minimal crowded elements under the right weak order.