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Cayley graphs of abelian groups and their quantum symmetries

Primarily AI-generated textHuman understanding: some partsmath.QA — Quantum Algebramath.OA — Operator Algebras

Contributed by Teo Banica ↗

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

Building on the recent work of Marashdeh on the circulant graphs of prime order, we study the quantum symmetries of the Cayley graphs of the finite abelian groups AA. Our main result states that if the coherent algebra is the orbital algebra of A⋊EA\rtimes E, with ∣E∣=k|E|=k and with the non-trivial elements of EE having at most mm fixed points, then the graph has no quantum symmetry when ∣A∣≥k2m|A|\geq k^2m. We also settle the case of the circulant graphs with pnp^n vertices.

Provenance statement

Written by Claude Fable, with help from Teo Banica.

Tools used

Anthropic
Claude Fable

References

  1. A. Atserias, L. Mančinska, D.E. Roberson, R. Šámal, S. Severini and A. Varvitsiotis, Quantum and non-signalling graph isomorphisms, J. Combin. Theory Ser. B 136 (2019), 289--328.
  2. T. Banica, Quantum automorphism groups of homogeneous graphs, J. Funct. Anal. 224 (2005), 243--280.
  3. T. Banica and J. Bichon, Free product formulae for quantum permutation groups, J. Inst. Math. Jussieu 6 (2007), 381--414.
  4. T. Banica and J. Bichon, Quantum automorphism groups of vertex-transitive graphs of order ≤11\leq11, J. Algebraic Combin. 26 (2007), 83--105.
  5. T. Banica, J. Bichon and G. Chenevier, Graphs having no quantum symmetry, Ann. Inst. Fourier 57 (2007), 955--971.
  6. T. Banica and J.P. McCarthy, The Frucht property in the quantum group setting, Glasg. Math. J. 64 (2022), 603--633.
  7. J. Bichon, Quantum automorphism groups of finite graphs, Proc. Amer. Math. Soc. 131 (2003), 665--673.
  8. J. Bichon, Free wreath product by the quantum permutation group, Algebr. Represent. Theory 7 (2004), 343--362.
  9. A. Chassaniol, Study of quantum symmetries for vertex-transitive graphs using intertwiner spaces, 1904.00455 .
  10. J. van Dobben de Bruyn, D.E. Roberson and S. Schmidt, Asymmetric graphs with quantum symmetry, Proc. Lond. Math. Soc. 131 (2025), e70098.
Show all 31 referencesShow fewer references
  1. E. Dobson, P. Spiga and G. Verret, Cayley graphs on abelian groups, Combinatorica 36 (2016), 371--393.
  2. S. Evdokimov and I. Ponomarenko, Characterization of cyclotomic schemes and normal Schur rings over a cyclic group, St. Petersburg Math. J. 14 (2003), 189--221.
  3. D. Gromada, Quantum symmetries of Cayley graphs of abelian groups, Glasg. Math. J. 65 (2023), 655--686.
  4. R. Haque, U. Karmakar and A. Mandal, Some results on Bichon's quantum automorphism group of graphs, 2511.15334 .
  5. K.H. Leung and S.H. Man, On Schur rings over cyclic groups, II, J. Algebra 183 (1996), 273--285.
  6. K.H. Leung and S.H. Man, On Schur rings over cyclic groups, Israel J. Math. 106 (1998), 251--267.
  7. M. Lupini, L. Mančinska and D.E. Roberson, Nonlocal games and quantum permutation groups, J. Funct. Anal. 279 (2020), 108592.
  8. M.F. Marashdeh, Two-basepoint Terwilliger algebras and the quantum symmetry of prime-order circulants, 2609.00462 .
  9. P. Meunier, Quantum properties of F\mathcal F-cographs, 2312.01516 .
  10. S.R. Mondal and J. Nisar, Quantum rigidity of Cayley graphs: an algebraic approach via Schur ring, Mathematics 14 (2026), 3513.
  11. B. Musto, D. Reutter and D. Verdon, The Morita theory of quantum graph isomorphisms, Comm. Math. Phys. 365 (2019), 797--845.
  12. M. Muzychuk and I. Ponomarenko, On pseudocyclic association schemes, Ars Math. Contemp. 5 (2012), 1--25.
  13. M. Muzychuk and I. Ponomarenko, On quasi-thin association schemes, J. Algebra 351 (2012), 467--489.
  14. J. Schanz, Quantum symmetries of combinatorial structures: a computer assisted approach, PhD Thesis, Saarland University (2025).
  15. S. Schmidt, The Petersen graph has no quantum symmetry, Bull. Lond. Math. Soc. 50 (2018), 395--400.
  16. S. Schmidt, Quantum automorphisms of folded cube graphs, Ann. Inst. Fourier 70 (2020), 949--970.
  17. S. Schmidt, On the quantum symmetry of distance-transitive graphs, Adv. Math. 368 (2020), 107150.
  18. S. Schmidt, Quantum automorphism groups of finite graphs, PhD Thesis, Saarland University (2020).
  19. S. Schmidt and M. Weber, Quantum symmetries of graph C∗C^*-algebras, Canad. Math. Bull. 61 (2018), 848--864.
  20. O. Verbitsky and M. Zhukovskii, Canonization of a random circulant graph by counting walks, Combin. Probab. Comput. (2025), 1--29.
  21. S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195--211.

Version history

  1. v1 · 30 pagesSubmitted by Teo BanicaInitial depositCurrentNov 11, 2026