Alcove path and Nichols-Woronowicz model of the equivariant KK-theory of generalized flag varieties

Fomin and Kirillov initiated a line of research into the realization of the cohomology and $K$-theory of generalized flag varieties $G/B$ as commutative subalgebras of certain noncommutative algebras. This approach has several advantages, which we discuss. This paper contains the most comprehensive result in a series of papers related to the mentioned line of research. More precisely, we give a model for the $T$-equivariant $K$-theory of a generalized flag variety $K_T(G/B)$ in terms of a certain braided Hopf algebra called the Nichols-Woronowicz algebra. Our model is based on the Chevalley-type multiplication formula for $K_T(G/B)$ due to the first author and Postnikov; this formula is stated using certain operators defined in terms of so-called alcove paths (and the corresponding affine Weyl group). Our model is derived using a type-independent and concise approach

Noncommutative algebras related with Schubert calculus on Coxeter groups

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

Lefschetz elements of Artinian Gorenstein algebras and Hessians of homogeneous polynomials (Representation Theory and Combinatorics)

This article is based on my joint work with Junzo Watanabe [8]. The Lef-schetz property is a ring-theoretic abstraction of the Hard Lefschetz Theorem for compact K\"ahler manifolds. The following are fundamental problems on the study of the Lefschetz property for Artinian graded algebras: Problem 0.1. For a given graded Artinian algebra $A$ , decide whether or not $A$ has the strong (or weak) Lefschetz property. Problem 0.2. When a graded Artinian algebra $A$ has the strong Lefschetz property, determine the set of Lefschetz elements in the part $A_{1}$ of degree one. In this work, we give a characterization of the Lefschetz elements in Ar-tinian Gorenstein rings over a field $k$ of characteristic zero in terms of the higher Hessians. As an application, we give new examples of Artinian Goren-stein rings which do not have the strong Lefschetz property