A Crystal on Decreasing Factorizations in the 0-Hecke Monoid

Abstract

We introduce a type $A$ crystal structure on decreasing factorizations of fully-commu\-tative 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.

Uncrowding algorithm for hook-valued tableaux

“…It was proved in [Buc02, Section 6] that U SVT in (3.1) is a bijection. Monical, Pechenik and Scrimshaw [MPS20] proved that U SVT intertwines with the crystal operators on set-valued tableaux (see also [MPPS20]). A similar uncrowding algorithm for multiset-valued tableaux was given in [HS20, Section 3.2].…”

“…An important property of the uncrowding algorithm on set-valued tableaux is that it intertwines with crystal operators [MPS20] (see also [MPPS20]). The crystal structure on a combinatorial set is the combinatorial shadow of a (quantum) group representation (see for example [HK02,BS17]).…”

Uncrowding Algorithm for Hook-Valued Tableaux

“…It was proved in [3,Section 6] that U SVT in (3.1) is a bijection. Monical et al [13] proved that U SVT intertwines with the crystal operators on set-valued tableaux (see also [12]). A similar uncrowding algorithm for multiset-valued tableaux was given in [8, Section 3.2].…”

“…An important property of the uncrowding algorithm on set-valued tableaux is that it intertwines with crystal operators [13] (see also [12]). The crystal structure on a combinatorial set is the combinatorial shadow of a (quantum) group representation (see, for example, [2,7]).…”

RSK Tableaux and the Weak Order on Fully Commutative Permutations

“…First introduced in [Ste96a], the fully commutative elements of a Coxeter group have the property that every pair of reduced words are related by a sequence of commutation relations. This set of objects is combinatorially rich and has been studied extensively (see, for example, [MPPS20,Nad15,Ste98]). A permutation is fully commutative if and only if it avoids the pattern 321 [BJS93], and the fully commutative permutations are exactly those with fewer than three rows in their Robinson-Schensted-Knuth (RSK) tableaux [Sch61].…”