Lock polynomials and lock tableaux are natural analogues to key polynomials and Kohnert tableaux, respectively. In this paper, we compare lock polynomials to the much-studied key polynomials and give an explicit description of a crystal structure on lock tableaux. Furthermore, we construct an injective, weight-preserving map from lock tableaux to Kohnert tableaux that intertwines with their respective crystal operators. As a result, we see that the crystal structure on lock tableaux has a natural embedding into the Demazure crystal. We also examine the conditions for which key and lock polynomials are symmetric or quasisymmetric.
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I am grateful to Sami Assaf for pointing me to this question and for the enlightening (and patient) conversations that followed and to Jim Haglund, Jongwon Kim, and Vasu Tewari for their support. I am also grateful to the referees for improving this paper with their numerous helpful suggestions. The author was partially supported by the NSF Graduate Research Fellowship, DGE-1845298.
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Supported by NSF DGE-1845298.
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Wang, G. Locks Fit into Keys: A Crystal Analysis of Lock Polynomials. Ann. Comb. 24, 767–789 (2020). https://doi.org/10.1007/s00026-020-00513-4