The A.B.C.Ds of Schubert calculus

TL;DR

This paper introduces Atiyah-Bott Combinatorial Dreams in Schubert calculus, establishing new relations between equivariant structure coefficients across different flag manifolds and exploring their implications in quantum cohomology.

Contribution

It presents novel combinatorial relations in Schubert calculus, linking equivariant structure coefficients of isotropic flag manifolds and extending understanding in quantum cohomology.

Findings

Relation between equivariant structure coefficients for two isotropic flag manifolds

Complementary results to a theorem in quantum cohomology

K-theory results rule out similar correspondences in K-theory

Abstract

We collect Atiyah-Bott Combinatorial Dreams (A.B.C.Ds) in Schubert calculus. One result relates equivariant structure coefficients for two isotropic flag manifolds, with consequences to the thesis of C. Monical. We contextualize using work of N. Bergeron-F. Sottile, S. Billey-M. Haiman, P. Pragacz, and T. Ikeda-L. Mihalcea-I. Naruse. The relation complements a theorem of A. Kresch-H. Tamvakis in quantum cohomology. Results of A. Buch-V. Ravikumar rule out a similar correspondence in K-theory.