Abstract
We suggest a new combinatorial construction for the co-homology ring of the flag manifold.
The degree 2 commutation relations satisfied by the divided difference operators corresponding to positive roots define a quadratic associative algebra. In this algebra, the formal analogues of Dunkl operators generate a commutative subring, which is shown to be canonically isomorphic to the cohomology of the flag manifold. This leads to yet another combinatorial version of the corresponding Schubert calculus.
The paper contains numerous conjectures and open problems. We also discuss a generalization of the main construction to quantum cohomology.
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Fomin, S., Kirillov, A.N. (1999). Quadratic Algebras, Dunkl Elements, and Schubert Calculus. In: Brylinski, JL., Brylinski, R., Nistor, V., Tsygan, B., Xu, P. (eds) Advances in Geometry. Progress in Mathematics, vol 172. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1770-1_8
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