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The foundation of generalized parallel connections, 2-sums, and segment-cosegment exchanges of matroids

Matthew Baker, Oliver Lorscheid, Zach Walsh, Tianyi Zhang

TL;DR

The paper develops a tensor-product framework for matroid foundations over pastures to analyze how representability behaves under key matroid operations. It proves that, when $M_1|T=M_2|T$ and $T$ is modular in both, the foundation of the generalized parallel connection satisfies $F_M\cong F_{M_1}\otimes_{F_T}F_{M_2}$, and specializes to the complete tensor product in the 2-sum case and preserves the foundation under segment-cosegment exchange. These foundational equalities induce bijections on rescaling classes of representations and preserve universal partial fields, connecting structural matroid operations with algebraic tensor products. The results yield new consequences for orientability/rigidity and excluded minor theory, expanding our understanding of representability across pastures and partial fields with broad implications for matroid representation theory.

Abstract

We show that, under suitable hypotheses, the foundation of a generalized parallel connection of matroids is the relative tensor product of the foundations. Using this result, we show that the foundation of a 2-sum of matroids is the absolute tensor product of the foundations, and that the foundation of a matroid is invariant under segment-cosegment exchange.

The foundation of generalized parallel connections, 2-sums, and segment-cosegment exchanges of matroids

TL;DR

The paper develops a tensor-product framework for matroid foundations over pastures to analyze how representability behaves under key matroid operations. It proves that, when and is modular in both, the foundation of the generalized parallel connection satisfies , and specializes to the complete tensor product in the 2-sum case and preserves the foundation under segment-cosegment exchange. These foundational equalities induce bijections on rescaling classes of representations and preserve universal partial fields, connecting structural matroid operations with algebraic tensor products. The results yield new consequences for orientability/rigidity and excluded minor theory, expanding our understanding of representability across pastures and partial fields with broad implications for matroid representation theory.

Abstract

We show that, under suitable hypotheses, the foundation of a generalized parallel connection of matroids is the relative tensor product of the foundations. Using this result, we show that the foundation of a 2-sum of matroids is the absolute tensor product of the foundations, and that the foundation of a matroid is invariant under segment-cosegment exchange.
Paper Structure (14 sections, 34 theorems, 79 equations, 1 figure)

This paper contains 14 sections, 34 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on foundations and representations of matroids over pastures
  3. Pastures
  4. Examples
  5. Tensor products
  6. Representations of matroids over pastures
  7. The universal pasture and the foundation
  8. Induced representations for embedded minors
  9. Cross ratios
  10. Facts about generalized parallel connections
  11. The foundation of a generalized parallel connection
  12. The foundation of a 2-sum
  13. The foundation of a segment-cosegment exchange
  14. Application to a conjecture by Pendavingh and van Zwam

Key Result

Theorem A

Let $M_1$ and $M_2$ be matroids so that $E(M_1) \cap E(M_2) = T$ and $M_1|T = M_2|T$. Suppose that either: Then the foundation of $P_T(M_1,M_2)$ is isomorphic to $F_{M_1}\otimes_{F_{M_1|T}}F_{M_2}$.

Figures (1)

  • Figure 1: A generalized parallel connection for which the foundation of $P_T(M_1, M_2)$ is not isomorphic to $F_{M_1} \otimes_{F_T} F_{M_2}$.

Theorems & Definitions (73)

  • Theorem A
  • Corollary 1
  • Remark
  • Theorem B
  • Theorem C
  • Remark
  • Corollary 2
  • Corollary 3
  • Definition
  • Corollary 4
  • ...and 63 more