  Lock polynomials and lock Kohnert tableaux are natural analogues to key
polynomials and key Kohnert tableaux, respectively. In this paper, we compare
lock polynomials to the much-studied key polynomials and show that the
difference of a key polynomial and lock polynomial for the same composition is
monomial positive. We also examine the conditions for which key and lock
polynomials are symmetric or quasisymmetric. We accomplish these goals
combinatorially using key Kohnert tableaux and lock Kohnert tableaux. In
particular, for the difference of a key minus a lock, we focus on the behavior
of crystal operators on Kohnert tableaux. The Type A Demazure crystal can be
realized on the vertex set of key Kohnert tableaux, and we show with an
explicit combinatorial definition that a similar crystal-like structure exists
on the vertex set of lock Kohnert tableaux. Finally, we construct an injective,
weight-preserving map from lock Kohnert tableaux to key Kohnert tableaux that
intertwines the crystal operators.

Combinatorics