## 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.