What Do You Know About Schubert Polynomials?

Schubert polynomials are generalizations of Schur polynomials, which are polynomials that represent homology classes. Homology classes are fundamental objects in algebraic topology that capture the properties and structure of spaces. Schubert polynomials provide a way to study and compute these homology classes in a more general setting. Therefore, the correct answer is homology classes.

Lascoux and Schutzenberger introduced Schubert polynomials.

Richard P. Stanley is the correct answer because he is known for his work in algebraic combinatorics and has made significant contributions to the field. In particular, he has put forth a conjectural rule for the coefficients of certain combinatorial polynomials, which suggests a relationship between various combinatorial objects. This conjecture has been influential in the study of combinatorial structures and has led to further research and developments in the field.

The history of Schubert polynomials was described in the year 1995.

Pipe dreams or rc-graphs are called the Schubert polynomials that can be seen as a generating function over certain combinatorial objects.

Fomin, Gelfand, and Postnikov introduced quantum Schubert polynomials in 1997.

Schubert polynomials have positive coefficients. This means that each term in the polynomial has a positive coefficient, indicating that the polynomial is always positive or non-negative.

Double Schubert polynomials are polynomials in two infinite sets of variables. These polynomials are used in algebraic geometry and combinatorics to study the intersection theory on Grassmannians and flag varieties. The two infinite sets of variables correspond to the Schubert cells and the opposite Schubert cells, which are important objects in the study of flag varieties. The double Schubert polynomials encode information about the intersection numbers of these cells and have applications in representation theory and algebraic geometry.

Lascoux is the correct answer because they are the one who described the history of Schubert polynomials.

Schubert polynomials are named after Hermanne Schubert. The spelling of her name is "Hermanne" and not "Hermaine" or "Herman" or "Henry". Therefore, the correct answer is Hermanne Schubert.