Lascoux polynomials provide a simultaneous generalization of several important families of polynomials arising in algebraic combinatorics: Schur polynomials, appearing as the characters of irreducible GL(n)-modules and as representatives of Schubert classes in the cohomology ring of a Grassmannian; key polynomials, defined as the characters of Demazure modules in GL(n)-modules; and, finally, symmetric Grothendieck polynomials, that geometrically correspond to the classes of structure sheaves of Schubert varieties in the K-ring of a Grassmannian.

I will speak about a new combinatorial presentation of Lascoux polynomials in terms of Gelfand-Zetlin polytopes. We show that they can be obtained from a certain explicitly described cellular decomposition of a Gelfand-Zetlin polytope as sums of monomials corresponding to the cells of this decomposition that belong to a given set of faces of the polytope. This interpretation for Schur polynomials provides the classical Weyl character formula, and for Demazure modules we recover a result by V.Kiritchenko, V.Timorin and myself (2012) on integer points in faces of Gelfand-Zetlin polytopes. The talk is based on our joint work with Ekaterina Presnova, arXiv: 2312.01417.