Equivariant Cohomology, Schubert Calculus, and Edge Labeled Tableaux
EQUIVARIANT COHOMOLOGY, SCHUBERT CALCULUS, AND EDGE LABELED TABLEAUX COLLEEN ROBICHAUX, HARSHIT YADAV, AND ALEXANDER YONG 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 combina- torics, 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. To William Fulton on his eightieth birthday, for inspiring generations. 1. INTRODUCTION 1.1. Purpose. Singular cohomology is a functor between the categories ftopological spaces; continuous mapsg ! fgraded algebras; homomorphismsg: The cohomology functor links the geometry of Grassmannians to symmetric functions and Young tableaux. However, this does not take into account the large torus action on the Grassmannian. A similar functor for topological spaces with continuous group actions is equivariant cohomology. What are equivariant analogues for these centerpieces of algebraic combinatorics? We posit a comprehensive answer, with applications, and future perspectives. n 1.2. Schubert calculus. Let X = Grk(C ) be the Grassmannian of k-dimensional planes n in C . The group GLn of invertible n×n matrices acts transitively on X by change of basis. Let B− ⊂ GLn be a opposite Borel subgroup of lower triangular matrices. B− acts on X ◦ with finitely many orbits Xλ where λ is a partition (identified with its Young diagram, in English notation) that is contained in the k × (n − k) rectangle Λ.
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