A Pieri rule for key polynomials
Schur functions are an amazing basis of symmetric functions originally defined as characters of irreducible modules for $GL_n$. The Pieri rule for the product of a Schur function and a single row Schur function is a multiplicity-free branching rule with a beautiful combinatorial interpretation in terms of adding boxes to a Young diagram. Key polynomials are an interesting basis of the polynomial ring originally defined as characters of submodules for irreducible $GL_n$ modules under the action of upper triangular matrices. In this talk, I'll present joint work with Danjoseph Quijada where we give a Pieri rule for the product of a key polynomial and a single row key polynomial. While this formula has signs, it is multiplicity-free and has an interpretation in terms of adding balls to a key diagram.