Lucas Gagnon: The quasisymmetric flag variety and equivariant forest polynomials

Lucas Gagnon, York University

Schubert calculus transforms the intersection theory of the flag variety GL_n/B into a multiplication problem for combinatorial polynomials, while double Schubert calculus extends this to a torus-equivariant setting. Two essential components underlie these approaches: (1) a surjective homomorphism from the polynomial ring F[x1,…,xn] to the (torus-equivariant) cohomology ring of GL_n/B, and (2) the existence of Schubert polynomials, a basis of F[x1,…,xn] that interacts naturally with the surjection from (1). This talk will describe a subvariety of GL_n/B that exhibits a surprisingly tight analogue of (1) and (2) with respect to another basis of combinatorial polynomials and generalize this basis to the equivariant setting.  I will show how these new objects can be analyzed using tools from algebraic combinatorics such as noncrossing partitions and quasisymmetric polynomials and speculate about how this can deepen our understanding of equivariant Schubert calculus. This is ongoing work with Nantel Bergeron, Philippe Nadeau, Hunter Spink, and Vasu Tewari.