TY - GEN

T1 - Existence of Quantum Symmetries for Graphs on Up to Seven Vertices

T2 - 2022 International Symposium on Symbolic and Algebraic Computation - ISSAC '22

AU - Levandovskyy, Viktor

AU - Eder, Christian

AU - Steenpass, Andreas

AU - Schmidt, Simon

AU - Schanz, Julien

AU - Weber, Moritz

PY - 2022

Y1 - 2022

N2 - The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In general, there are more quantum symmetries than symmetries and it is a non-trivial task to determine when this is the case for a given graph: The question is whether or not the associative algebra associated to the quantum automorphism group is commutative. We use noncommutative Gröbner bases in order to tackle this problem; the implementation uses Gap and Singular:Letterplace. We determine the existence of quantum symmetries for all connected, undirected graphs without multiple edges and without self-edges, for up to seven vertices. As an outcome, we infer within our regime that a classical automorphism group of order one or two is an obstruction for the existence of quantum symmetries.

AB - The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In general, there are more quantum symmetries than symmetries and it is a non-trivial task to determine when this is the case for a given graph: The question is whether or not the associative algebra associated to the quantum automorphism group is commutative. We use noncommutative Gröbner bases in order to tackle this problem; the implementation uses Gap and Singular:Letterplace. We determine the existence of quantum symmetries for all connected, undirected graphs without multiple edges and without self-edges, for up to seven vertices. As an outcome, we infer within our regime that a classical automorphism group of order one or two is an obstruction for the existence of quantum symmetries.

U2 - 10.1145/3476446.3535481

DO - 10.1145/3476446.3535481

M3 - Article in proceedings

SP - 311

EP - 318

BT - ISSAC '22: Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

PB - ACM Association for Computing Machinery

Y2 - 4 July 2022 through 7 July 2022

ER -