Quadratic Algebras, Dunkl Elements, and Schubert Calculus
نویسندگان
چکیده
We suggest a new combinatorial construction for the cohomology ring of the ag manifold. The degree 2 commutation relations satissed by the divided diierence operators corresponding to positive roots deene a quadratic associative algebra. In this algebra, the formal analogues of Dunkl operators generate a commuta-tive subring, which is shown to be canonically isomorphic to the cohomology of the ag 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. Contents
منابع مشابه
On some quadratic algebras
We study some quadratic algebras which are appeared in the low–dimensional topology and Schubert calculus. We introduce the Jucys–Murphy elements in the braid algebra and in the pure braid group, as well as the Dunkl elements in the extended affine braid group. Relationships between the Dunkl elements, Dunkl operators and Jucys–Murphy elements are described.
متن کاملOn a Quantum Version of Pieri's Formula
We give an algebro-combinatorial proof of a general version of Pieri’s formula following the approach developed by Fomin and Kirillov in the paper “Quadratic algebras, Dunkl elements, and Schubert calculus.” We prove several conjectures posed in their paper. As a consequence, a new proof of classical Pieri’s formula for cohomology of complex flag manifolds, and that of its analogue for quantum ...
متن کاملBraided differential structure on Weyl groups, quadratic algebras and elliptic functions
We discuss a class of generalized divided difference operators which give rise to a representation of Nichols-Woronowicz algebras associated to Weyl groups. For the root system of type A, we also study the condition for the deformations of the Fomin-Kirillov quadratic algebra, which is a quadratic lift of the Nichols-Woronowicz algebra, to admit a representation given by generalized divided dif...
متن کاملFibered Quadratic Hopf Algebras Related to Schubert Calculus
We introduce and study certain quadratic Hopf algebras related to Schu-bert calculus of the ag manifold.
متن کاملNoncommutative algebras related with Schubert calculus on Coxeter groups
For any finite Coxeter system (W,S) we construct a certain noncommutative algebra, the so-called bracket algebra, together with a family of commuting elements, the so-called Dunkl elements. The Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the Coxeter group W. We prove this conjecture for classical Coxeter groups and I2(m). We def...
متن کامل