Quantum automorphisms of matroids

Abstract

Motivated by the vast literature of quantum automorphism groups of graphs, we define and study quantum automorphism groups of matroids. A key feature of quantum groups is that there are many quantizations of a classical group, and this phenomenon manifests in the cryptomorphic characterizations of matroids. Our primary goal is to understand the resulting structures from an algebraic and computational point of view. In particular, we investigate the relationship between these quantum groups and to find when these quantum groups exhibit quantum symmetry. Finally, we prove a matroidal analog of Lovász's theorem characterizing graph isomorphisms in terms of homomorphism counts.