Existence of Quantum Symmetries for Graphs on Up to Seven Vertices: A Computer based Approach
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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.
Original language | English |
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Title of host publication | ISSAC '22: Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation |
Publisher | ACM Association for Computing Machinery |
Publication date | 2022 |
Pages | 311-318 |
DOIs | |
Publication status | Published - 2022 |
Event | 2022 International Symposium on Symbolic and Algebraic Computation - ISSAC '22 - Villeneuve-d'Ascq, France Duration: 4 Jul 2022 → 7 Jul 2022 |
Conference
Conference | 2022 International Symposium on Symbolic and Algebraic Computation - ISSAC '22 |
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Land | France |
By | Villeneuve-d'Ascq |
Periode | 04/07/2022 → 07/07/2022 |
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