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Lock polynomials and lock Kohnert tableaux are natural analogues to keypolynomials and key Kohnert tableaux, respectively. In this paper, we comparelock polynomials to the much-studied key polynomials and show that thedifference of a key polynomial and lock polynomial for the same composition ismonomial positive. We also examine the conditions for which key and lockpolynomials are symmetric or quasisymmetric. We accomplish these goalscombinatorially using key Kohnert tableaux and lock Kohnert tableaux. Inparticular, for the difference of a key minus a lock, we focus on the behaviorof crystal operators on Kohnert tableaux. The Type A Demazure crystal can berealized on the vertex set of key Kohnert tableaux, and we show with anexplicit combinatorial definition that a similar crystal-like structure existson the vertex set of lock Kohnert tableaux. Finally, we construct an injective,weight-preserving map from lock Kohnert tableaux to key Kohnert tableaux thatintertwines the crystal operators.

Locks fit into keys: a crystal analysis of lock polynomials