Quadratic algebras, Dunkl elements, and Schubert calculus

A. Kirillov

CRM/St. Petersburg

We define an associative algebra whose generators are labelled by the positive roots and are subject to certain quadratic relations. In this algebra, the formal analogues of Dunkl operators generate a commutative ring which happens to be cannonically isomorphic to the cohomology ring of the flag variety. This leads to yet another combinatorial approach to the Schubert calculus. We also discuss a generalisation of the main construction to the quantum cohomology and quantum Schubert calculus.