Recipe theorem for the Tutte polynomial for matroids, renormalization group-like approach

TL;DR

This paper introduces a novel proof of the recipe theorem for the Tutte polynomial of matroids using a quantum field theory-inspired renormalization group approach, linking Hopf algebra characters to the polynomial.

Contribution

It provides a new Hopf algebraic method to prove the recipe theorem and related convolution formula for the Tutte polynomial of matroids.

Findings

New proof of the recipe theorem for matroid Tutte polynomial

Hopf algebra characters relate to the Tutte polynomial

Alternative proof of the convolution formula in matroid theory

Abstract

Using a quantum field theory renormalization group-like differential equation, we give a new proof of the recipe theorem for the Tutte polynomial for matroids. The solution of such an equation is in fact given by some appropriate characters of the Hopf algebra of isomorphic classes of matroids, characters which are then related to the Tutte polynomial for matroids. This Hopf algebraic approach also allows to prove, in a new way, a matroid Tutte polynomial convolution formula appearing in W. Kook {\it et. al., J. Comb. Series} {\bf B 76} (1999).