Equivariant cohomology, Schubert calculus, and edge labeled tableaux

TL;DR

This paper surveys the use of edge labeled Young tableaux in modeling equivariant Schubert calculus, introduces a new shifted analogue, and discusses conjectures linking combinatorics with geometric representation theory.

Contribution

It introduces a shifted analogue of edge labeled tableaux and proposes a conjectural Littlewood-Richardson rule for the Anderson-Fulton ring related to equivariant cohomology.

Findings

Survey of combinatorial and geometric results

Introduction of shifted edge labeled tableaux

Conjecture on Littlewood-Richardson rule for Anderson-Fulton ring

Abstract

This chapter concerns edge labeled Young tableaux, introduced by H. Thomas and the third author. It is used to model equivariant Schubert calculus of Grassmannians. We survey results, problems, conjectures, together with their influences from combinatorics, algebraic and symplectic geometry, linear algebra, and computational complexity. We report on a new shifted analogue of edge labeled tableaux. Conjecturally, this gives a Littlewood-Richardson rule for the structure constants of the D. Anderson-W. Fulton ring, which is related to the equivariant cohomology of isotropic Grassmannians.