Matt Samuel,  (part 1) "A Molev-Sagan type formula for double Schubert polynomials"
Channel: Schubert Seminar
Date: 08/10/2023

Abstract: We introduce Schubert polynomials and double Schubert polynomials, which share their structure constants with the cohomology rings and torus-equivariant cohomology rings of the complete flag variety, respectively, in the basis of Schubert classes. We discuss known positivity results about their structure constants, including positive formulas in certain special cases. We discuss divided difference operators, skew divided difference operators, and their relationship to Schubert polynomials and their structure constants. Then we consider the Molev-Sagan case, where we multiply double Schubert polynomials in two different sets of coefficient variables. We detail a positivity conjecture for the resulting coefficients, and present a new Pieri formula for multiplying a double Schubert polynomial by a factorial elementary symmetric polynomial as well as a factorial complete symmetric polynomial, both in a different set of variables. We then show how to use the Pieri formula for factorial elementary symmetric polynomials to multiply a large class of pairs of double Schubert polynomials in different sets of variables, specifically the case of separated descents, which provides previously unknown generalizations to the Molev-Sagan case as well as to equivariant cohomology of several known results about multiplying ordinary Schubert polynomials, the latest of which was published in 2019. We then discuss how the Pieri formula can be used to separate the variables in a double Schubert polynomial, opening up avenues to computing more coefficients.