A Puzzle Formula for H*T x Cx(T*Pn)

Voula Collins

Séminaire lotharingien de combinatoire, 78B (2017)

Résumé

We will begin with the work of Davesh Maulik and Andrei Okounkov where they define a "stable basis" for the T-equivariant cohomology ring H^*_{T x C^x}(T^*Gr_k(Cⁿ)), of the cotangent bundle to a Grassmannian. Just as we can compute the product structure of the the cohomology ring of a Grassmannian using Schubert classes as a basis, it is natural to attempt to do the same for the cotangent bundle to a Grassmannian using these Maulik-Okounkov classes as a basis. In this paper I compute the structure constants of both the regular and equivariant cohomology rings of the cotangent bundle to projective space, using Maulik-Okounkov classes as a basis. First I do so directly in Theorem 3.1, and then I put forth a conjectural positive formula, which uses a variant of Knutson-Tao puzzles, in Conjecture 4.2. The proof of the puzzle formula relies on an explicit rational function identity that I have checked through dimension 9.