Equivariant $K$-theory of cellular toric varieties

TL;DR

This paper characterizes the equivariant topological K-theory of cellular toric varieties, showing it is isomorphic to piecewise Laurent polynomial functions on the fan, and provides explicit bases and structure constants.

Contribution

It introduces a description of the equivariant K-ring for cellular toric varieties, establishing an isomorphism with piecewise Laurent polynomials and computing explicit bases and structure constants.

Findings

K-theory ring is isomorphic to piecewise Laurent polynomial functions

Explicit basis for the K-theory as a module over the representation ring

Computed multiplicative structure constants for the basis

Abstract

In this article we describe the $T_{comp}$-equivariant topological $K$-ring of a $T$-{\it cellular} complete toric variety. We further show that $K_{T_{comp}}^0(X)$ is isomorphic as an $R(T_{comp})$-algebra to the ring of piecewise Laurent polynomial functions on the associated fan denoted $PLP(\Delta)$. Furthermore, we compute a basis for $K_{T_{comp}}^0(X)$ as a $R(T_{comp})$-module and multiplicative structure constants with respect to this basis.