More Like this

1. 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. 

   more » « less
2. Crystal for Stable Grothendieck Polynomials

   Morse, Jennifer ; Pan, Jianping ; Poh, Wencin ; Schilling, Anne ( April 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. 

   more » « less
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. 

   more » « less
4. On the Okounkov–Olshanski formula for standard tableaux of skew shapes

   Morales, A. H. ; Zhu, D. ( July 2020 , Séminaire lotharingien de combinatoire)

   The classical hook-length formula counts the number of standard tableaux of straight shapes, but there is no known product formula for skew shapes. Okounkov– Olshanski (1996) and Naruse (2014) found new positive formulas for the number of standard Young tableaux of a skew shape. We prove various properties of the Okounkov– Olshanski formula: a reformulation similar to the Naruse formula, determinantal formulas for the number of terms, and a q-analogue extending the formula to reverse plane partitions, which complements work by Chen and Stanley for semistandard tableaux. 

   more » « less
5. Combinatorics of Generalized Exponents

   https://doi.org/10.1093/imrn/rny157  

   Lecouvey, Cédric ; Lenart, Cristian ( July 2018 , International Mathematics Research Notices)

   Abstract We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type $A_{n-1}$, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type $C_{n}$, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type $A_{2n-1}$, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig $t$-analogs associated to zero-weight spaces in the irreducible representations of symplectic Lie algebras. We also present three applications of our combinatorial formula and discuss some implications to relating two type $C$ branching rules. Our methods are expected to extend to the orthogonal types. 

   more » « less