Positivity of Peterson Schubert Calculus

Abstract

The Peterson variety is a subvariety of the flag manifold G/B equipped with an action of a one-dimensional torus, and a torus invariant paving by affine cells, called Peterson cells. We prove that the equivariant pull-backs of Schubert classes indexed by certain Coxeter elements are dual (up to an intersection multiplicity) to the fundamental classes of Peterson cell closures. Dividing these classes by the intersection multiplicities yields a Z-basis for the equivariant cohomology of the Peterson variety. We prove several properties of this basis, including a Graham positivity property for its structure constants, and stability with respect to inclusion in a larger Peterson variety. We also find formulae for intersection multiplicities with Peterson classes. This explains geometrically, in arbitrary Lie type, recent positivity statements proved in type A by Goldin and Gorbutt.

Equivariant Chevalley, Giambelli, and Monk Formulae for the Peterson Variety

“…The multiplicities m(v I ), which depend only on I and not on ∆, were calculated for certain Coxeter elements v I in [GMS21, Thm 1.3]. We prove in Theorem 5.7 a general formula for m(v I ), for any Coxeter element v I , conjectured in [GMS21].…”

“…The Peterson variety admits a natural C * action. In [GMS21], Goldin, Mihalcea, and Singh show that C * -equivariant Peterson Schubert calculus also satisfies Graham positivity; see Equation (1) below. The equivariant cohomology of Peterson varieties in all Lie types is described in [HHM15], using generators and relations.…”

A filtration on the cohomology rings of regular nilpotent Hessenberg varieties

“…Abe, Horiguchi, Kuwata, and Zeng, give a geometric interpretation in the context of ordinary cohomology [4] and they also give a different formula for the structure constants by introducing a combinatorial object called left-right diagrams and defining a combinatorial game using these diagrams. In addition, recent work of Goldin, Mihalcea, and Singh gives a geometric interpretation to the equivariant Peterson Schubert classes and the Graham-positivity of their products, in arbitrary Lie type [32]. Finally, follow-up work of Goldin and Singh [33] derives, again in arbitrary Lie type, new explicit formulas for Monk and Chevalley rules in H * S (P et) using modified degree-2 classes that are different from those used by Drellich.…”

The cohomology rings of intersections of Peterson varieties and Richardson varieties

“…The Peterson variety has been much studied from several view points (e.g. [3,8,12,13,18,19,22]). An explicit presentation of the cohomology ring 1 of Pet Φ given in [13] is uniform across Lie types, as explained below.…”