RSK tableaux and the weak order on fully commutative permutations

TL;DR

This paper explores the structure of fully commutative permutations using RSK tableaux, introducing boolean cores and classifying permutations into crowded and uncrowded, revealing their order dynamics and minimal crowded elements.

Contribution

It introduces boolean cores for fully commutative permutations and characterizes the uncrowded permutations via RSK tableaux, advancing understanding of their order structure.

Findings

Uncrowded permutations share RSK tableaux with their boolean cores.

Minimal crowded elements are characterized by permutation patterns and descents.

The dynamics of the weak order reveal when permutations transition between crowded and uncrowded.

Abstract

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.