#### IMSc Webinar

#### Pieri Rules for Polynomials

#### Sami Assaf

##### University of Southern California

*Webinar link: us02web.zoom.us/j/88554180523*

Schur functions are an amazing basis of symmetric functions originally defined as characters of irreducible modules for of 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 joint work with Danjoseph Quijada, 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, perhaps after dropping some balls down. Time permitting, I’ll give applications to Schubert polynomials where the signs cancel to give a positive Pieri formula.

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