Extended quadratic algebra and a model of the equivariant cohomology ring of flag varieties

TL;DR

This paper introduces an extended quadratic algebra model for the equivariant cohomology ring of flag varieties, generalizing existing algebraic structures and providing new formulas for Schubert calculus.

Contribution

It constructs an extended quadratic algebra for type A and extends Nichols-Woronowicz algebras to model equivariant cohomology of flag varieties for general Coxeter systems.

Findings

Extended quadratic algebra models type A flag variety cohomology

Generalized equivariant Pieri rule for double Schubert polynomials

Extended Nichols-Woronowicz algebra for crystallographic Coxeter systems

Abstract

For the root system of type $A$ we introduce and study a certain extension of the quadratic algebra invented by S. Fomin and the first author, to construct a model for the equivariant cohomology ring of the corresponding flag variety. As an application of our construction we describe a generalization of the equivariant Pieri rule for double Schubert polynomials. For a general finite Coxeter system we construct an extension of the corresponding Nichols-Woronowicz algebra. In the case of finite crystallographic Coxeter systems we present a construction of extended Nichols-Woronowicz algebra model for the equivariant cohomology of the corresponding flag variety.