Algebra and valuations related to the Tutte polynomial

TL;DR

This chapter explores algebraic structures and valuation theories related to the Tutte polynomial, including Orlik-Solomon algebras, evaluative functions on matroid polytopes, and Hopf-algebra frameworks.

Contribution

It provides a comprehensive overview of algebraic and valuation approaches to the Tutte polynomial, integrating recent developments in Hopf algebras and G-invariant evaluations.

Findings

Introduction to Orlik-Solomon algebras

Sketch of evaluative functions on matroid polytopes

Discussion of Hopf-algebra structures in Tutte theory

Abstract

This is a chapter destined for the book "Handbook of the Tutte Polynomial". The chapter is a composite. The first part is a brief introduction to Orlik-Solomon algebras. The second part sketches the theory of evaluative functions on matroid base polytopes and in particular, the G-invariant (as the subject is known late 2015). A third very short section is on Hopf-algebra or coalgebra structures in Tutte polynomial theory.