Multiplicity-Free Key Polynomials

The key polynomials, defined by Lascoux–Schützenberger, are characters for the Demazure modules of type A. We classify multiplicity-free key polynomials. The proof uses two combinatorial models for key polynomials. The first is due to Kohnert. The second is in terms of Searles’ rule for the quasi-key polynomials of Assaf–Searles. Our argument proves a sufficient condition for a quasi-key polynomial to be multiplicity-free.