Lusztig showed that all polynomials $p(x_{1,1}, \dotsc,x_{n,n})$ in the dual canonical basis satisfy $p(A) \geq 0$ for every totally nonnegative matrix $A = (a_{i,j})$. It is also possible to show that the evaluation of these polynomials at Jacobi-Trudi matrices yields Schur-nonnegative symmetric functions. We will discuss variations of these properties and their connection to Schubert varieties and cluster algebras.