Equivariant map

About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.

An equivariant Riemann-Roch theorem for complete, simplicial toric varieties.

Abstract: Introduction. The theory of toric varieties establishes a now classical connection between algebraic geometry and convex polytopes. In particular, äs observed by Danilov in the seventies, finding a closed formula for the Todd class of complete toric varieties would have important consequences for enumeration of lattice points in convex lattice polytopes. Since then, a number of such formulas have been proposed; see [M], [Pl], [P2] The Todd class of complete simplicial toric varieties is computed in [G-G-K], using the Riemann-Roch formula of T. Kawasaki [Ka].

Affine Weyl Groups in K-Theory and Representation Theory

Abstract: We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P which is a direct generalization of the classical Chevalley formula. Our formula implies a simple combinatorial model for the characters of the irreducible representations of G and, more generally, for the Demazure characters. This model can be viewed as a discrete counterpart of the Littelmann path model, and has several advantages. Our construction is given in terms of a certain R-matrix, that is, a collection of operators satisfying the Yang-Baxter equation. It reduces to combinatorics of decompositions in the affine Weyl group and enumeration of saturated chains in the Bruhat order on the (nonaffine) Weyl group. Our model easily implies several symmetries of the coefficients in the Chevalley-type formula. We also derive a simple formula for multiplying an arbitrary Schubert class by a divisor class, as well as a dual Chevalley-type formula. The paper contains other applications and examples.

Theory of Degrees with Applications to Bifurcations and Differential Equations

Abstract: Elements of Differential Topology. Degree in Finite--Dimensional Spaces. Leray--Schauder Degree for Compact Fields. Nussbaum--Sadovskii Degree for Condensing Fields. Applications to Bifurcation Theory. S 1 --Equivariant Degree. Global Hopf Bifurcation Theory. Equivariant Degree of Dold--Ulrich. References. Index.

Quantum difference equation for Nakajima varieties

Abstract: For an arbitrary Nakajima quiver variety $X$, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the fundamental groupoid of a certain periodic locally finite hyperplane arrangement in $Pic(X)\otimes {\mathbb{C}}$. We identify the lattice part of this groupoid with the operators of quantum difference equation for $X$. The cases of quivers of finite and affine type are illustrated by explicit examples.