Renormalization group-like proof of the universality of the Tutte polynomial for matroids

TL;DR

This paper presents a new proof of the universality of the Tutte polynomial for matroids using Hopf algebra characters and differential equations inspired by renormalization group flow, offering a novel algebraic perspective.

Contribution

It introduces a Hopf algebra-based approach to prove Tutte polynomial universality and provides an alternative proof of a known convolution formula for matroid Tutte polynomials.

Findings

Hopf algebra characters satisfy differential equations similar to renormalization group equations

New proof of Tutte polynomial universality for matroids

Alternative proof of the matroid Tutte polynomial convolution formula

Abstract

In this paper we give a new proof of the universality of the Tutte polynomial for matroids. This proof uses appropriate characters of Hopf algebra of matroids, algebra introduced by Schmitt (1994). We show that these Hopf algebra characters are solutions of some differential equations which are of the same type as the differential equations used to describe the renormalization group flow in quantum field theory. This approach allows us to also prove, in a different way, a matroid Tutte polynomial convolution formula published by Kook, Reiner and Stanton (1999). This FPSAC contribution is an extended abstract.