# On volume functions of special flow polytopes

## TL;DR

This paper investigates the volume of specific flow polytopes, demonstrating it satisfies differential equations and providing an inductive formula related to root system rank, advancing understanding of their geometric properties.

## Contribution

It establishes a differential equation framework for flow polytope volumes and derives an inductive formula based on root system rank, offering new computational tools.

## Key findings

- Volume satisfies a unique differential system
- Inductive formula for volume based on root system rank
- Provides a new approach to computing flow polytope volumes

## Abstract

In this paper, we consider the volume of a special kind of flow polytope. We show that its volume satisfies a certain system of differential equations, and conversely, the solution of the system of differential equations is unique up to a constant multiple. In addition, we give an inductive formula for the volume with respect to the rank of the root system of type A.