Asymptotics of Symmetry in Matroids

TL;DR

This paper demonstrates that as the size of matroids grows, most have very simple symmetry groups, either trivial or generated by a single transposition, with a focus on sparse paving matroids.

Contribution

It establishes the asymptotic behavior of automorphism groups in matroids, revealing that most have minimal symmetry, especially among sparse paving matroids.

Findings

Almost all matroids have trivial automorphism groups.

Most sparse paving matroids have trivial automorphism groups.

Automorphism groups are typically very simple in large matroids.

Abstract

We prove that asymptotically almost all matroids have a trivial automorphism group, or an automorphism group generated by a single transposition. Additionally, we show that asymptotically almost all sparse paving matroids have a trivial automorphism group.