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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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Uncrowding Algorithm for Hook-Valued Tableaux
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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- PAR ID:
- 10325534
- Date Published:
- Journal Name:
- Annals of Combinatorics
- Volume:
- 26
- Issue:
- 1
- ISSN:
- 0218-0006
- Page Range / eLocation ID:
- 261 to 301
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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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
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00026-022-00567-6
