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Chain Tutte polynomials

Max Wakefield

TL;DR

This work introduces chain Tutte polynomials T^k_M as a refined spectrum between the classical Tutte polynomial and Derksen's G-invariant, built from iterated convolutions of universal Tutte characters. It proves these polynomials are valuative on generalized permutahedra and establishes a generalized deletion-contraction rule, yielding a versatile recursion framework. The paper connects T^k_M to several known matroid invariants, including the Möbius polynomial, opposite characteristic polynomial, generalized Möbius polynomial, and Ford's S-polynomial, and shows how Derksen's G-invariant is recovered in the top chain case, hence situating T^k_M as a bridge between foundational invariants. The approach provides a computational and conceptual toolkit for accessing finer matroid invariants and paves the way for systematic evaluations of new polynomials against classical invariants. The results have potential implications for understanding valuative invariants, matroid polytopes, and realization spaces through a unifying chain-polynomial perspective.

Abstract

The Tutte polynomial and Derksen's $\mathcal{G}$-invariant are the universal deletion-contraction and valuative matroid and polymatroid invariants, respectively. There are only a handful of well known invariants (like the matroid Kazhdan-Lusztig polynomials) between (in terms of fineness) the Tutte polynomial and Derksen's $\mathcal{G}$-invariant. The aim of this study is to define a spectrum of generalized Tutte polynomials to fill the gap between the Tutte polynomial and Derksen's $\mathcal{G}$-invariant. These polynomials are built by taking repeated convolution products of universal Tutte characters studied by Dupont, Fink, and Moci and using the framework of Ardila and Sanchez for studying valuative invariants. We develop foundational aspects of these polynomials by showing they are valuative on generalized permutahedra and present a generalized deletion-contraction formula. We apply these results on chain Tutte polynomials to obtain formulas for the Möbius polynomial, the opposite characteristic polynomial, a generalized Möbius polynomial, Ford's expected codimension of a matroid variety, and Derksen's $\mathcal{G}$-invariant.

Chain Tutte polynomials

TL;DR

This work introduces chain Tutte polynomials T^k_M as a refined spectrum between the classical Tutte polynomial and Derksen's G-invariant, built from iterated convolutions of universal Tutte characters. It proves these polynomials are valuative on generalized permutahedra and establishes a generalized deletion-contraction rule, yielding a versatile recursion framework. The paper connects T^k_M to several known matroid invariants, including the Möbius polynomial, opposite characteristic polynomial, generalized Möbius polynomial, and Ford's S-polynomial, and shows how Derksen's G-invariant is recovered in the top chain case, hence situating T^k_M as a bridge between foundational invariants. The approach provides a computational and conceptual toolkit for accessing finer matroid invariants and paves the way for systematic evaluations of new polynomials against classical invariants. The results have potential implications for understanding valuative invariants, matroid polytopes, and realization spaces through a unifying chain-polynomial perspective.

Abstract

The Tutte polynomial and Derksen's -invariant are the universal deletion-contraction and valuative matroid and polymatroid invariants, respectively. There are only a handful of well known invariants (like the matroid Kazhdan-Lusztig polynomials) between (in terms of fineness) the Tutte polynomial and Derksen's -invariant. The aim of this study is to define a spectrum of generalized Tutte polynomials to fill the gap between the Tutte polynomial and Derksen's -invariant. These polynomials are built by taking repeated convolution products of universal Tutte characters studied by Dupont, Fink, and Moci and using the framework of Ardila and Sanchez for studying valuative invariants. We develop foundational aspects of these polynomials by showing they are valuative on generalized permutahedra and present a generalized deletion-contraction formula. We apply these results on chain Tutte polynomials to obtain formulas for the Möbius polynomial, the opposite characteristic polynomial, a generalized Möbius polynomial, Ford's expected codimension of a matroid variety, and Derksen's -invariant.
Paper Structure (20 sections, 29 theorems, 116 equations, 2 figures, 2 tables)

This paper contains 20 sections, 29 theorems, 116 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Posets and polymatroids
  4. Möbius function
  5. Characteristic polynomials and deletion-contraction
  6. Basic properties of chain Tutte polynomials
  7. Formulas for $T^k$ in terms of $T^{|\mathcal{A}|}$
  8. Derksen's $G$-invariant
  9. Direct sums and duals
  10. Basic Examples
  11. Recursion for chain Tutte polynomials
  12. Generalized permutahedra and Valuations
  13. Generalized permutahedra
  14. Matroid valuations
  15. Chain Tutte evaluations and specializations
  16. ...and 5 more sections

Key Result

Theorem 1.3

Let $M=(\mathcal{A},\mathrm{rk})$ be a matroid and $a\in \mathcal{A}$.

Figures (2)

  • Figure 1: The Gray graphs
  • Figure 2: Two non-isomorphic matroids with same Tutte polynomial $T^1$ but different chain Tutte polynomial $T^2$

Theorems & Definitions (66)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1: Zas87, Proposition 7.1.4
  • Lemma 2.2
  • proof
  • Definition 2.3
  • ...and 56 more