Combinatorics Seminar: "Positivity results for Peterson Schubert calculus in all types"

Speaker: Rebecca Goldin, George Mason University

Abstract: Motivated by positivity in equivariant Schubert calculus, we introduce a set of bases for the equivariant cohomology of the Peterson variety P, a subvariety of G/B.  We will explicitly describe these bases with a focus on their algebraic and combinatorial properties.  In this talk, I will present results about these bases, and their associated “Peterson Schubert calculus”.   First, we show that, for each basis,  the expansion of a product of elements results in structure constants (coefficients of the product, after expanding in the basis) that are polynomials in one variable with nonnegative coefficients, a remarkable phenomenon reflecting properties of the affine paving of Peterson varieties.  Indeed, each basis satisfies a notion of duality with a set of smaller Peterson varieties, under a pairing between cohomology and homology.  

Though the structure constants are nonnegative, there are no combinatorial (positive) formulas to describe them. We will present a Monk formula in all types, providing an explicit combinatorial formula for the product of a Peterson Schubert divisor with any other Peterson Schubert class, and a dual Chevalley formula. These results, rather strangely, boil down to a relationship between the non-equivariant homology class of the Peterson variety and that of the permutahedron.

Finally, for Peterson Schubert calculus in type A we have an explicit combinatorial formula for all structure constants (in a particular basis), which was spoken about recently in the WUSL combinatorics seminar. Such formulas give us hope for similar success with other types and other Hessenberg varieties.  
This work is separately joint with Leonardo Mihalcea (Virginia Tech), Rahul Singh (ICERM), and Brent Gorbutt (GMU).


Host: Martha Precup