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arXiv · 2510.04336

Combinatorial Aspects of Elliptic Schubert Calculus

Cristian Lenart·Rui Xiong·Changlong Zhong
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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.

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Cristian Lenart, Rui Xiong, Changlong Zhong. 2025-10-05. Combinatorial Aspects of Elliptic Schubert Calculus. https://arxiv.org/abs/2510.04336

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