Computations on the tautological basis of the cohomology ring of the Peterson variety

TL;DR

This paper investigates the algebraic structure of the cohomology ring of the Peterson variety, providing explicit formulas for product expansions of certain classes, which enhances understanding of its combinatorial and geometric properties.

Contribution

It introduces a new explicit expansion formula for products of degree 2 classes in the cohomology ring of the Peterson variety, expressed via elementary symmetric polynomials and binomial coefficients.

Findings

Derived a square free expansion formula for products of degree 2 classes.

Connected the product expansions to elementary symmetric polynomials.

Provided combinatorial formulas for cohomology class multiplications.

Abstract

It is known that the set of square free monomials on the Chern classes of the tautological line bundles over the Peterson variety forms an additive basis of its cohomology ring. We study the expansion formula for their products. In particular, we give a square free expansion of the products multiplying degree 2 classes in terms of elementary symmetric polynomials and binomial coefficients.