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

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.

Host: Martha Precup