Schubert polynomials

In this thesis, Schubert polynomials and their types, which are the basic tools of the Schubert Calculus (Calculus), are studied. The properties of classical Schubert, double Schubert, rational Schubert, quantum Schubert polynomials are compiled and exemplified using Lascoux (2013), Macdonald(1991), and Billey and Haiman(1994) studies. In addition, the relations we have obtained between rational Schubert polynomials and elementary symmetric polynomials are given, and a closed formula is given for quantum elementary symmetric functions that have an important place for quantum Schubert polynomials. With this formula, the quantum multiplier Y_u(q1 , . . . , qn−2 ) of the double Schubert polynomial Y_u(x;y) for the maximum permutation ω_0 ∈ S_n is defined and its combinatorial interpretation is given. Where q_1 , . . . , q_n−2 are variables and x = {x_1 , . . . , x_n }, y = {y_1 , . . . , y_n } and u is the code of ω_0 . It is proved that the double quantum Schubert polynomial is the product of the double Schubert polynomial Y_u(x;y) and the quantum multiplier. The definition of the Y_u^q,rat(x;y) quantum-rational Schubert polynomial is given. With the help of this definition, K_u^q,rat(x) quantum-rational Key and G_u^q,rat(x;y) quantum-rational Grothendieck polynomials are expressed.