The Quasisymmetric Flag Variety

In this talk I will construct a “quasisymmetric flag variety”, a subvariety of the complete type A  flag variety built by adapting the BGG geometric construction of divided differences to the newly introduced “quasisymmetric divided differences” (introduced by Vasu Tewari in a previous talk). Its cohomology ring is the ring of quasisymmetric coinvariants, and it is paved by certain Weyl group translates of “Coxeter Richardson varieties” which are Kronecker dual to the basis of forest polynomials.

From studying the torus-equivariant geometry of this variety we discover a new notion of “equivariant quasisymmetry”, which leads to a new family of “double fundamental quasisymmetric polynomials” and “double forest polynomials” that have Graham-positive multiplication structure constants.

Joint with Nantel Bergeron, Lucas Gagnon, Philippe Nadeau, Vasu Tewari