Combinatorial operators on polynomials

Alain Lascoux

The symmetric group (and more generally, the classical Weyl groups) can be used to compute in the ring of polynomials in a finite number of variables. There are several actions of the symmetric group on this ring, we shall consider those which are deformations of the Newton divided differences. They all satisfy the braid relations and the Yang-Baxter equations.

This gives simple and explicit linear bases of the ring of polynomials, such as Schubert polynomials, Grothendieck polynomials, Demazure characters, Macdonald polynomials, each of these families extending the family of Schur symmetric functions, and presenting similar algebraic and combinatorial properties. After defining these bases, and giving some of their main properties, we shall illustrate how to use them for computing in several variables.