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1. Crystal for Stable Grothendieck Polynomials

   Morse, Jennifer; Pan, Jianping; Poh, Wencin; Schilling, Anne (January 2020, Séminaire lotharingien de combinatoire)

   We introduce a type A crystal structure on decreasing factorizations on 321-avoiding elements in the 0-Hecke monoid which we call *-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 *-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. 

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2. A Crystal on Decreasing Factorizations in the 0-Hecke Monoid

   https://doi.org/10.37236/9168  

   Morse, Jennifer; Pan, Jianping; Poh, Wencin; Schilling, Anne (April 2020, The Electronic Journal of Combinatorics)

   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. 

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3. Uncrowding Algorithm for Hook-Valued Tableaux

   https://doi.org/10.1007/s00026-022-00567-6  

   Pan, Jianping; Pappe, Joseph; Poh, Wencin; Schilling, Anne (March 2022, Annals of Combinatorics)

   Abstract Whereas set-valued tableaux are the combinatorial objects associated to stable Grothendieck polynomials, hook-valued tableaux are associated with 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. 

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4. 0-Hecke modules for row-strict dual immaculate functions

   https://doi.org/10.1090/tran/9006  

   Niese, Elizabeth; Sundaram, Sheila; van_Willigenburg, Stephanie; Vega, Julianne; Wang, Shiyun (February 2024, Transactions of the American Mathematical Society)

   We introduce a new basis of quasisymmetric functions, the row-strict dual immaculate functions. We construct a cyclic, indecomposable 0-Hecke algebra module for these functions. Our row-strict immaculate functions are related to the dual immaculate functions of Berg-Bergeron-Saliola-Serrano-Zabrocki (2014-15) by the involution $\mathrm{\psi <\#comment/>}$on the ring $\mathrm{QSym}$of quasisymmetric functions. We give an explicit description of the effect of $\mathrm{\psi <\#comment/>}$on the associated 0-Hecke modules, via the poset induced by the 0-Hecke action on standard immaculate tableaux. This remarkable poset reveals other 0-Hecke submodules and quotient modules, often cyclic and indecomposable, notably for a row-strict analogue of the extended Schur functions studied in Assaf-Searles (2019). Like the dual immaculate function, the row-strict dual immaculate function is the generating function of a suitable set of tableaux, corresponding to a specific descent set. We give a complete combinatorial and representation-theoretic picture by constructing 0-Hecke modules for the remaining variations on descent sets, and showing thatallthe possible variations for generating functions of tableaux occur as characteristics of the 0-Hecke modules determined by these descent sets. 

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5. Arithmetic fundamental lemma for the spherical Hecke algebra

   https://doi.org/10.1007/s00229-024-01572-0  

   Li, Chao; Rapoport, Michael; Zhang, Wei (June 2024, manuscripta mathematica)

   Abstract We define Hecke correspondences and Hecke operators on unitary RZ spaces and study their basic geometric properties, including a commutativity conjecture on Hecke operators. Then we formulate the arithmetic fundamental lemma conjecture for the spherical Hecke algebra. We also formulate a conjecture on the abundance of spherical Hecke functions with identically vanishing first derivative of orbital integrals. We prove these conjectures for the case$$\textrm{U} (1)\times \textrm{U} (2)$$ $\text{U}\left(1\right)\times \text{U}\left(2\right)$. 

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