Purity and Separation for Oriented Matroids / Pavel Galashin and Alexander Postnikov.

Format:

Book

Author/Creator:

Galashin, Pavel, author.

Postnikov, Alexander, author.

Series:

Memoirs of the American Mathematical Society ; Volume 289.

Memoirs of the American Mathematical Society Series ; Volume 289

Language:

English

Subjects (All):

Combinatorial analysis.

Physical Description:

1 online resource (92 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

Leclerc and Zelevinsky, motivated by the study of quasi-commuting quantum flag minors, introduced the notions of strongly separated and weakly separated collections. These notions are closely related to the theory of cluster algebras, to the combinatorics of the double Bruhat cells, and to the totally positive Grassmannian. A key feature, called the purity phenomenon, is that every maximal by inclusion strongly (resp., weakly) separated collection of subsets in [n] has the same cardinality. In this paper, we extend these notions and define M-separated collections for any oriented matroid M. We show that maximal by size M-separated collections are in bijection with fine zonotopal tilings (if M is a realizable oriented matroid), or with one-element liftings of M in general position (for an arbitrary oriented matroid). We introduce the class of pure oriented matroids for which the purity phenomenon holds: an oriented matroid M is pure if M-separated collections form a pure simplicial complex, i.e., any maximal by inclusion M-separated collection is also maximal by size. We pay closer attention to several special classes of oriented matroids: oriented matroids of rank 3, graphical oriented matroids, and uniform oriented matroids. We classify pure oriented matroids in these cases. An oriented matroid of rank 3 is pure if and only if it is a positroid (up to reorienting and relabeling its ground set). A graphical oriented matroid is pure if and only if its underlying graph is an outerplanar graph, that is, a subgraph of a triangulation of an n-gon. We give a simple conjectural characterization of pure oriented matroids by forbidden minors and prove it for the above classes of matroids (rank 3, graphical, uniform).

Contents:

Cover

Title page

Chapter 1. Introduction

Acknowledgments

Chapter 2. Separation, purity, and zonotopal tilings

2.1. Separation for vector configurations

2.2. Separation for oriented matroids

2.3. Zonotopal tilings

2.4. Pure oriented matroids

2.5. Mutation-closed domains

Chapter 3. Motivating examples

3.1. Alternating oriented matroids

3.2. Strong separation

3.3. Chord separation

3.4. Weak separation

Chapter 4. Simple operations on oriented matroids

4.1. Relabeling and adding/removing loops and coloops

4.2. Adding/removing parallel elements

4.3. Reorientations

Chapter 5. Main results on purity

5.1. Purity of matroids of rank 2 or corank 1

5.2. Purity of rank 3 oriented matroids

5.3. Graphical oriented matroids

5.4. Uniform oriented matroids

5.5. Arbitrary oriented matroids

Chapter 6. Background on zonotopal tilings and oriented matroids

6.1. Sets and signed vectors

6.2. Zonotopal tilings

6.3. Oriented matroids

Chapter 7. Maximal by size ℳ-separated collections

7.1. Mutation-closed domains

7.2. The structure of the alternating matroid of corank 2

7.3. Colocalizations and complete collections

7.4. Proof of Theorem 7.2

7.5. Zonotopal tilings for oriented matroids

Chapter 8. Pure oriented matroids

8.1. Proof of Lemma 8.2

Chapter 9. The graphical case

9.1. Pure graphs

9.2. Mutation-closed domains for graphical oriented matroids

9.3. Enumerating maximal -separated collections

9.4. Regular matroids

Chapter 10. The rank 3 case

Chapter 11. Classification results

11.1. Oriented matroids of rank 4 and corank 2

Bibliography

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

Other Format:

Print version: Galashin, Pavel Purity and Separation for Oriented Matroids

ISBN:

1-4704-7594-4