A positive formula for type $A$ Peterson Schubert calculus

Abstract

Peterson varieties are special nilpotent Hessenberg varieties that have appeared in the study of quantum cohomology, representation theory, and combinatorics. In type $A$, the Peterson variety $Y$ is a subvariety of the complete flag variety $Fl(n;\mathbb{C})$, and is invariant under the action of a subgroup $S\cong {\mathbb{C}}^{\ast }$ of $T$, where $T$ is the standard (noncompact) torus acting on $Fl(n;\mathbb{C})$. Using the Peterson Schubert basis introduced by Harada and Tymoczko obtained by restricting a specific set of Schubert classes from $H_T^{\ast }(Fl(n;\mathbb{C}))$ to $H_S^{\ast }(Y)$, we describe the product structure of the equivariant cohomology $H_S^{\ast }(Y)$. In particular, we show that the product is manifestly positive in an appropriate sense by providing an explicit positive combinatorial formula for its structure constants. Our method requires a new combinatorial identity of binomial coefficients that generalizes Vandermonde's identity.

48 pages, significant edits for clarity