A Positive Formula for Type A Peterson Schubert Calculus

Abstract

The Peterson variety is a special case of a nilpotent Hessenberg variety, a class of subvarieties of G/B that have appeared in the study of quantum cohomology, representation theory and combinatorics. In type A, the Peterson variety Y is a subvariety of $$Fl(n;{\mathbb {C}})$$ , the set of complete flags in $${\mathbb {C}}^n$$ , and comes equipped with an action by a one-dimensional torus subgroup S of a standard torus T that acts on $$Fl(n;{\mathbb {C}})$$ . Using the Peterson Schubert basis introduced in Harada and Tymoczko (Proc Lond Math Soc 103(1):40–72, 2011) and obtained by restricting a specific set of Schubert classes from $$H_T^*(Fl(n; {\mathbb {C}}))$$ to $$H_S^*(Y)$$ , we describe the product structure of the equivariant cohomology $$H_{S}^*(Y)$$ . 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. A key step in our proof requires a new combinatorial identity of binomial coefficients that generalizes Vandermonde’s identity, and merits independent interest.