Existence of Quantum Symmetries for Graphs on Up to Seven Vertices: A Computer based Approach
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Existence of Quantum Symmetries for Graphs on Up to Seven Vertices : A Computer based Approach. / Levandovskyy, Viktor; Eder, Christian; Steenpass, Andreas; Schmidt, Simon; Schanz, Julien; Weber, Moritz.
ISSAC '22: Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation. ACM Association for Computing Machinery, 2022. s. 311-318.Publikation: Bidrag til bog/antologi/rapport › Konferencebidrag i proceedings › Forskning › fagfællebedømt
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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 -
ID: 312696088