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.

锁多项式和锁tableaux分别是密钥多项式和Kohnert tableaux的自然类似物。在本文中，我们将锁定多项式与经过广泛研究的密钥多项式进行了比较，并给出了锁定平台上晶体结构的明确描述。此外，我们构造了一个从锁tableaux到Kohnert tableaux的内射式，保重地图，并与它们各自的晶体算子交织在一起。结果，我们发现锁屏上的晶体结构自然嵌入了Demazure晶体中。我们还研究了密钥和锁多项式对称或准对称的条件。