A crystal on decreasing factorizations in the 00-Hecke monoid

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.Comment: 37 pages; in revision 1 Sections 3.1, 4.3 and 4.4 were updated; in revision 2 the phrase 321-avoiding is replaced by fully-commutative, typos are fixed, reference adde

Uncrowding algorithm for hook-valued tableaux

Whereas set-valued tableaux are the combinatorial objects associated to stable Grothendieck polynomials, hook-valued tableaux are associated to stable canonical Grothendieck polynomials. In this paper, we define a novel uncrowding algorithm for hook-valued tableaux. The algorithm ``uncrowds'' the entries in the arm of the hooks and yields a set-valued tableau and a column-flagged increasing tableau. We prove that our uncrowding algorithm intertwines with crystal operators. An alternative uncrowding algorithm that ``uncrowds'' the entries in the leg instead of the arm of the hooks is also given. As an application of uncrowding, we obtain various expansions of the canonical Grothendieck polynomials.Comment: 32 page

A trilogy of (super)crystals: the queer, the set-valued and the hook-valued

This dissertation compiles three main results concerning crystals or supercrystals. Firstly, we present a characterization for queer supercrystals introduced by Grantcharov et al. We also construct a type A crystal whose character is the stable Grothendieck polynomials for fully-commutative permutations. Finally, we describe an uncrowding map on hook-valued tableaux which intertwines the crystal operators on hook-valued tableaux to those of set-valued tableaux

A trilogy of (super)crystals: the queer, the set-valued and the hook-valued

This dissertation compiles three main results concerning crystals or supercrystals. Firstly, we present a characterization for queer supercrystals introduced by Grantcharov et al. We also construct a type A crystal whose character is the stable Grothendieck polynomials for fully-commutative permutations. Finally, we describe an uncrowding map on hook-valued tableaux which intertwines the crystal operators on hook-valued tableaux to those of set-valued tableaux

Uncrowding algorithm for hook-valued tableaux

Whereas set-valued tableaux are the combinatorial objects associated to stable Grothendieck polynomials, hook-valued tableaux are associated to stable canonical Grothendieck polynomials. In this paper, we define a novel uncrowding algorithm for hook-valued tableaux. The algorithm "uncrowds" the entries in the arm of the hooks and yields a set-valued tableau and a column-flagged increasing tableau. We prove that our uncrowding algorithm intertwines with crystal operators. An alternative uncrowding algorithm that "uncrowds" the entries in the leg instead of the arm of the hooks is also given. As an application of uncrowding, we obtain various expansions of the canonical Grothendieck polynomials