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Matroids arising from algebraic shifting

Lazar Guterman, Eran Nevo

TL;DR

The paperCharacterizes when the inverse image under exterior shifting of shifted $k$-uniform hypergraphs forms the bases of a matroid, showing this occurs precisely when the hyperedges constitute an initial lex-segment; it ties this to known matroids such as the Crapo-Rota simplicial matroid, hyperconnectivity, and area-rigidity, and extends the result to $k\ge4$ under a combinatorial condition. It also establishes a parallel theory for symmetric shifting, notably for graphs, where the generic rigidity matroid arises, and develops a Lex-matroidal framework that strengthens the connection between combinatorial structure and matroidality. The results blend Kalai-style shifting with determinant-based matroid representations (compound matrices) and Betti-number invariants, yielding both exact characterizations in low rank and conditional insights in higher rank, along with several open problems on extensions and rigidity interpretations.

Abstract

We characterize the shifted simple graphs and the $3$-uniform shifted hypergraphs whose inverse image under exterior shifting is the set of bases of a matroid: those are exactly the hypergraphs whose hyperedges form an initial lex-segment. There are several examples of known matroids arising in this way: the simplicial matroid, the hyperconnectivity matroid and the area-rigidity matroid. For $k\ge 4$, we provide a similar characterization for shifted $k$-uniform hypergraphs satisfying an additional combinatorial condition. For symmetric shifting, we prove an analogous characterization for shifted simple graphs, where the classical generic rigidity matroid is an example of a matroid arising in this way.

Matroids arising from algebraic shifting

TL;DR

The paperCharacterizes when the inverse image under exterior shifting of shifted -uniform hypergraphs forms the bases of a matroid, showing this occurs precisely when the hyperedges constitute an initial lex-segment; it ties this to known matroids such as the Crapo-Rota simplicial matroid, hyperconnectivity, and area-rigidity, and extends the result to under a combinatorial condition. It also establishes a parallel theory for symmetric shifting, notably for graphs, where the generic rigidity matroid arises, and develops a Lex-matroidal framework that strengthens the connection between combinatorial structure and matroidality. The results blend Kalai-style shifting with determinant-based matroid representations (compound matrices) and Betti-number invariants, yielding both exact characterizations in low rank and conditional insights in higher rank, along with several open problems on extensions and rigidity interpretations.

Abstract

We characterize the shifted simple graphs and the -uniform shifted hypergraphs whose inverse image under exterior shifting is the set of bases of a matroid: those are exactly the hypergraphs whose hyperedges form an initial lex-segment. There are several examples of known matroids arising in this way: the simplicial matroid, the hyperconnectivity matroid and the area-rigidity matroid. For , we provide a similar characterization for shifted -uniform hypergraphs satisfying an additional combinatorial condition. For symmetric shifting, we prove an analogous characterization for shifted simple graphs, where the classical generic rigidity matroid is an example of a matroid arising in this way.

Paper Structure

This paper contains 11 sections, 16 theorems, 38 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hypergraphs and simplicial complexes
  4. Algebraic shifting
  5. Matroids
  6. Matroids arising from exterior shifting
  7. Preparatory lemmas
  8. Proofs of main results
  9. Matroids arising from symmetric shifting
  10. Lex-matroidal hypergraphs
  11. Concluding remarks

Key Result

Theorem 1.2

For $k=2,3$, a shifted $k$-uniform hypergraph $H\subseteq\binom{[n]}{k}$ is matroidal if and only if the (hyper)edges of $H$ form an initial lex-segment.

Figures (4)

  • Figure 1: The graphs $H_1$ and $H_2$ violate the exchange axiom
  • Figure 2: Compound matrix $X((H\setminus e_1)\cup e_2;(H\setminus e_1)\cup S)$ and diagonal monomial.
  • Figure 3: Choice of $\{1ji\}$.
  • Figure 6: Matrix $Y(B,H)$.

Theorems & Definitions (45)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 3.1
  • Remark 3.2
  • Corollary 3.3
  • ...and 35 more