TL;DR
This paper proves the affine Pieri rule for the cohomology of the affine flag variety, confirming a conjecture and connecting cap operators with Pieri operators, advancing understanding in algebraic geometry and representation theory.
Contribution
It establishes the affine Pieri rule by linking cap operators on the affine nilHecke ring with known Pieri operators, confirming a conjecture in the field.
Findings
Proved the affine Pieri rule for the affine flag variety.
Connected cap operators with Pieri operators from prior work.
Validated the conjecture by Lam, Lapointe, Morse, and Shimozono.
Abstract
We prove the affine Pieri rule for the cohomology of the affine flag variety conjectured by Lam, Lapointe, Morse and Shimozono. We study the cap operator on the affine nilHecke ring that is motivated by Kostant and Kumar's work on the equivariant cohomology of the affine flag variety. We show that the cap operators for Pieri elements are the same as Pieri operators defined by Berg, Saliola and Serrano. This establishes the affine Pieri rule.
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