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

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.