Combinatorial Aspects of Elliptic Schubert Calculus
Cristian Lenart, Rui Xiong, Changlong Zhong
TL;DR
The paper advances elliptic Schubert calculus by extending two central combinatorial results—Billey-type localization and pipe dream polynomials—to equivariant elliptic cohomology. It builds a framework using elliptic Demazure–Lusztig operators, twisted group algebras, and dynamical parameters $\lambda$ and $\hbar$, then derives a universal elliptic Billey formula and a parabolic variant. A type $A$ diagrammatic approach via wiring diagrams and a pipe dream model with elliptic weights yields explicit polynomial representatives, while a 3D mirror symmetry perspective interrelates dual structures and confirms consistency through Yang–Baxter-type relations. The work also clarifies the degeneration to $K$-theory and situates partial flag varieties within this broader elliptic setting, providing practical computational tools for elliptic Schubert classes.
Abstract
The main goal of this paper is to extend two fundamental combinatorial results in Schubert calculus on flag manifolds from equivariant cohomology and $K$-theory to equivariant elliptic cohomology. The foundations of elliptic Schubert calculus were laid in a few relatively recent papers by Rimányi, Weber, and Kumar. They include the recursive construction of elliptic Schubert classes via generalizations of the cohomology and $K$-theory push-pull operators and the study of the corresponding Demazure algebra. We derive a Billey-type formula for the localization of elliptic Schubert classes (for partial flag manifolds of arbitrary type) and a pipe dream model for their polynomial representatives in the case of type $A$ flag manifolds. The latter extends the pipe dream model for double Schubert and Grothendieck polynomials. We also study the degeneration of elliptic Schubert classes to $K$-theory, which recovers the corresponding classical formulas.