A Pieri rule for key polynomials
Schur functions are an amazing basis of symmetric functions originally defined as characters of irreducible modules for GLn. 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 GLn 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.