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Publication# Equivariant K-theory of GKM bundles

Abstract

Given a fiber bundle of GKM spaces, pi: M -> B, we analyze the structure of the equivariant K-ring of M as a module over the equivariant K-ring of B by translating the fiber bundle, pi, into a fiber bundle of GKM graphs and constructing, by combinatorial techniques, a basis of this module consisting of K-classes which are invariant under the natural holonomy action on the K-ring of M of the fundamental group of the GKM graph of B. We also discuss the implications of this result for fiber bundles pi: M -> B where M and B are generalized partial flag varieties and show how our GKM description of the equivariant K-ring of a homogeneous GKM space is related to the Kostant-Kumar description of this ring.

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Related publications (1)

Related concepts (5)

Pi

The number pi (paɪ; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number pi appears in many formulae across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern.

Combinatorics

Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas.

Fiber bundle

In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map, that in small regions of behaves just like a projection from corresponding regions of to The map called the projection or submersion of the bundle, is regarded as part of the structure of the bundle.

Let T be a torus and B a compact T-manifold. Goresky et al. show in [3] that if B is (what was subsequently called) a GKM manifold, then there exists a simple combinatorial description of the equivari