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In this lecture from the Institute for Advanced Study's Special Year Seminar, Vasu Tewari from the University of Pennsylvania explores quasisymmetric divided differences and forest polynomials. Discover Postnikov's divided symmetrization and its remarkable "positivity" properties, best understood through quasisymmetric divided differences operators. Learn about a new basis of the polynomial ring adapted to these operators, similar to how ordinary divided differences interact with Schubert polynomials. Understand how this basis works with reduction modulo the ideal of positive degree quasisymmetric polynomials and how Schubert polynomials expand non-negatively in this basis—encoding Schubert class expansions of certain toric Richardson varieties. Follow the combinatorial procedure for computing these Schubert structure constants and explore connections to mixed Eulerian numbers and lattice point counts of permutahedra. This talk complements a follow-up presentation by Hunter Spink on the underlying geometry and represents joint work with Philippe Nadeau (Lyon) and Hunter Spink (Toronto).
