Infinite type toric varieties and Voronoi Tilings

TL;DR

This paper explores infinite type toric varieties derived from loop group representation theory, linking them to Voronoi tilings and extending classical toric geometry concepts.

Contribution

It introduces new examples of infinite type toric varieties based on Voronoi tilings and explains their relation to torus orbit closures in flag varieties.

Findings

Constructed examples from loop group representations.

Connected Voronoi tilings to infinite toric varieties.

Extended classical toric geometry concepts.

Abstract

An infinite type toric variety is a normal toric variety given by a fan with infinitely many cones. We construct examples in this paper coming from representation theory of loop groups. The fans that appear are cones on Voronoi tilings on a vector space equipped with an inner product. We also explain the affine analogue of the connection between a generic torus orbit closure in a flag variety and the closure a maximal torus in the wonderful compactification.