2020/04/13 by Goldin, Rebecca, Gorbutt, Brent · 1 citation
#05E15 (Secondary) #14M15 (Primary) 05A10 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2004.05959
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≅ \mathbb C^* 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 HT^*(Fl(n; \mathbb C)) to HS^*(Y), we describe the product structure of the equivariant cohomology HS^*(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.