A crystal on decreasing factorizations in the $0$-Hecke monoid

TL;DR

This paper introduces a new crystal structure called $ ext{\star}$-crystal on decreasing factorizations in the 0-Hecke monoid, connecting it with set-valued tableaux and defining a novel insertion algorithm that interacts with crystal operators.

Contribution

It develops a $K$-theoretic crystal structure on decreasing factorizations in the 0-Hecke monoid and introduces a new insertion algorithm linking these factorizations with semistandard Young tableaux.

Findings

The $ ext{\star}$-crystal intertwines with the crystal on set-valued tableaux.

The new insertion algorithm relates to Hecke insertion and uncrowding.

The crystal structure generalizes previous work on symmetric groups.

Abstract

We introduce a type $A$ crystal structure on decreasing factorizations of fully-commutative elements in the 0-Hecke monoid which we call $\star$-crystal. This crystal is a $K$-theoretic generalization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the $\star$-crystal intertwines with the crystal on set-valued tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorization to pairs of semistandard Young tableaux and prove several properties, such as its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators.