Noncommutative algebras related with Schubert calculus on Coxeter groups

Abstract

For any finite Coxeter system $(W,S)$ we construct a certain noncommutative algebra, so-called {\it bracket algebra}, together with a familiy of commuting elements, so-called {\it Dunkl elements.} Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the group $W.$ We prove this conjecture for classical Coxeter groups and $I_2(m)$. We define a ``quantization'' and a multiparameter deformation of our construction and show that for Lie groups of classical type and $G_2,$ the algebra generated by Dunkl elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define {\it quantum Bruhat representation} of the corresponding bracket algebra. We study in more detail relations and structure of $B_n$-, $D_n$- and $G_2$-bracket algebras, and as an application, discover {\it Pieri type formula} in the $B_n$-bracket algebra. As a corollary, we obtain Pieri type formula for multiplication of arbitrary $B_n$-Schubert classes by some special ones. Our Pieri type formula is a generalization of Pieri's formulas obtained by A. Lascoux and M.-P. Sch\"utzenberger for flag varieties of type $A.$ We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements which describes ``noncommutative differential geometry on a finite Coxeter group'' in a sense of S. Majid.