# Coxeter submodular functions and deformations of Coxeter permutahedra

## TL;DR

This paper characterizes the deformations of Coxeter permutahedra, extending the connection between submodular functions and generalized permutahedra to all finite reflection groups, with implications for combinatorial and geometric structures.

## Contribution

It provides a comprehensive description of the deformation cone of Coxeter permutahedra, generalizing known results to all finite reflection groups.

## Key findings

- Describes the nef cone of the associated toric variety.
- Includes polyhedral models for Coxeter-theoretic analogs.
- Extends the correspondence between submodular functions and generalized permutahedra.

## Abstract

We describe the cone of deformations of a Coxeter permutahedron, or equivalently, the nef cone of the toric variety associated to a Coxeter complex. This family of polytopes contains polyhedral models for the Coxeter-theoretic analogs of compositions, graphs, matroids, posets, and associahedra. Our description extends the known correspondence between generalized permutahedra, polymatroids, and submodular functions to any finite reflection group.