#####
Séminaire Lotharingien de Combinatoire, 78B.67 (2017), 12 pp.

# Voula Collins

# A Puzzle Formula for H^{*}_{T x
Cx}(*T*^{*}**P**^{n})

**Abstract.**
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 Cx}(*T*^{*}Gr_{k}(**C**^{n})), 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.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

The following versions are available: