HEXAGONbeta

Higher invariants of complex Hadamard matrices

Primarily AI-generated textHuman understanding: some partsmath.OA — Operator Algebrasmath.QA — Quantum Algebra

Contributed by Teo Banica ↗

Version 1 / Nov 13, 2026 / CC BY 4.0

Abstract

Associated to any complex Hadamard matrix H∈MN(T)H\in M_N(\mathbb T) is a quantum permutation group G⊂uSN+G\subset_uS_N^+, the Hopf image of the representation associated to the magic matrix formed by the rank 1 projections Pij=Proj(Hi/Hj)P_{ij}=Proj(H_i/H_j), and its invariants ck=dim⁡(Fix(u⊗k))c_k=\dim(Fix(u^{\otimes k})) are the dimensions of the relative commutants of the subfactor associated to HH. We compute these invariants well beyond the levels k=2,3k=2,3 previously known, up to k=9k=9 in the best cases, by using Jones--Wenzl and annular decompositions. For the Di\c t\u a deformations of the Fourier matrices we compute GG for all values of the parameters, and we study the order 8 families F8,5F_{8,5}, in connection with free wreath products. For the matrices of small order with c2=2c_2=2, such as the Karlsson ones at generic parameters, we find ck=Catkc_k=Cat_k at all the levels computed. Finally, via an integrality principle and character sums, we prove that the Paley II matrices, and more generally those coming from symmetric conference matrices, have infinite depth, as well as the Paley I matrices with 11≤q<10511\leq q<10^5.

Provenance statement

Written by Claude Opus, with help from Teo Banica.

Tools used

Anthropic
Claude Opus

References

  1. N. Andruskiewitsch and J. Devoto, Extensions of Hopf algebras, St. Petersburg Math. J. 7 (1996), 17--52.
  2. T. Banica, Representations of compact quantum groups and subfactors, J. Reine Angew. Math. 509 (1999), 167--198.
  3. T. Banica, Symmetries of a generic coaction, Math. Ann. 314 (1999), 763--780.
  4. T. Banica, Compact Kac algebras and commuting squares, J. Funct. Anal. 176 (2000), 80--99.
  5. T. Banica, Quantum automorphism groups of homogeneous graphs, J. Funct. Anal. 224 (2005), 243--280.
  6. T. Banica, The defect of generalized Fourier matrices, Linear Algebra Appl. 438 (2013), 3667--3688.
  7. T. Banica, Truncation and duality results for Hopf image algebras, Bull. Pol. Acad. Sci. Math. 62 (2014), 161--179.
  8. T. Banica, Invitation to Hadamard matrices, 1910.06911 (2024).
  9. T. Banica and J. Bichon, Free product formulae for quantum permutation groups, J. Inst. Math. Jussieu 6 (2007), 381--414.
  10. T. Banica and J. Bichon, Quantum groups acting on 4 points, J. Reine Angew. Math. 626 (2009), 75--114.
Show all 95 referencesShow fewer references
  1. T. Banica and J. Bichon, Hopf images and inner faithful representations, Glasg. Math. J. 52 (2010), 677--703.
  2. T. Banica and J. Bichon, Random walk questions for linear quantum groups, Int. Math. Res. Not. 2015 (2015), no. 24, 13406--13436.
  3. T. Banica, J. Bichon and B. Collins, The hyperoctahedral quantum group, J. Ramanujan Math. Soc. 22 (2007), 345--384.
  4. T. Banica, J. Bichon and J.-M. Schlenker, Representations of quantum permutation algebras, J. Funct. Anal. 257 (2009), 2864--2910.
  5. T. Banica and B. Collins, Integration over the Pauli quantum group, J. Geom. Phys. 58 (2008), 942--961.
  6. T. Banica and R. Speicher, Liberation of orthogonal Lie groups, Adv. Math. 222 (2009), 1461--1501.
  7. K. Beauchamp and R. Nicoara, Orthogonal maximal abelian ∗*-subalgebras of the 6×66\times 6 matrices, Linear Algebra Appl. 428 (2008), 1833--1853.
  8. P. Biane, Some properties of crossings and partitions, Discrete Math. 175 (1997), 41--53.
  9. J. Bichon, Free wreath product by the quantum permutation group, Algebr. Represent. Theory 7 (2004), 343--362.
  10. J. Bichon, Algebraic quantum permutation groups, Asian-Eur. J. Math. 1 (2008), 1--13.
  11. J. Bichon, Quotients and Hopf images of a smash coproduct, Tsukuba J. Math. 39 (2015), 285--310.
  12. D. Bisch and U. Haagerup, Composition of subfactors: new examples of infinite depth subfactors, Ann. Sci. École Norm. Sup. 29 (1996), 329--383.
  13. D. Bisch and V. Jones, Algebras associated to intermediate subfactors, Invent. Math. 128 (1997), 89--157.
  14. D. Bisch, V. Jones and Z. Liu, Singly generated planar algebras of small dimension, Part III, Trans. Amer. Math. Soc. 369 (2017), 2461--2476.
  15. G. Björck and R. Fröberg, A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic nn-roots, J. Symbolic Comput. 12 (1991), 329--336.
  16. M. Brannan, Reduced operator algebras of trace-preserving quantum automorphism groups, Doc. Math. 18 (2013), 1349--1402.
  17. M. Brannan, A. Chirvasitu and A. Freslon, Topological generation and matrix models for quantum reflection groups, Adv. Math. 363 (2020), 106982.
  18. M. Brannan, D. Gromada, R. Hernández Palomares and N. Priebe, Complex Hadamard matrices: quantum symmetries, equivalence and non-local games, 2608.14858 .
  19. R.D. Burstein, Group-type subfactors and Hadamard matrices, Trans. Amer. Math. Soc. 367 (2015), 6783--6807.
  20. W. Camp and R. Nicoara, Subfactors and Hadamard matrices, 0704.1128 .
  21. M. Cárdenes Wuttig and J. Tindall, A complete classification of complex Hadamard matrices of order six, 2608.18053 (2026).
  22. O. Chterental and D.Ž. oković, On orthostochastic, unistochastic and qustochastic matrices, Linear Algebra Appl. 428 (2008), 1178--1201.
  23. Claude and Teo, On the maximality conjecture for quantum permutation groups, preprint 2026, zenodo:23858973 (https://zenodo.org/records/23858973).link
  24. P. Diţă, Some results on the parametrization of complex Hadamard matrices, J. Phys. A 37 (2004), 5355--5374.
  25. J. van Dobben de Bruyn, A. Freslon, P.N. Kar, D.E. Roberson and P. Zeman, Free inhomogeneous wreath product of quantum groups, 2504.13826 .
  26. J. van Dobben de Bruyn, P.N. Kar, D.E. Roberson, S. Schmidt and P. Zeman, Quantum automorphism groups of trees, J. Noncommut. Geom. 20 (2026), 489--505.
  27. D. Enders, A characterization of semiprojectivity for subhomogeneous C∗C^*-algebras, Doc. Math. 21 (2016), 987--1049.
  28. P. Fima and A. Troupel, Generalized free wreath products and their operator algebras, 2504.00596 .
  29. I. Francis, R. Nicoara and C. Worley, Bismash product commuting squares, preprint.
  30. A. Freslon, Free wreath products with amalgamation, Comm. Algebra 51 (2023), 72--94.
  31. GPT and Teo, Regular complex Hadamard matrices of order 7,8, preprint 2026, zenodo:23934849 (https://zenodo.org/records/23934849).link
  32. J. Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), 77--101.
  33. D. Gromada, Quantum symmetries of Hadamard matrices, Trans. Amer. Math. Soc. 377 (2024), 6341--6377.
  34. P. Grossman and V.F.R. Jones, Intermediate subfactors with no extra structure, J. Amer. Math. Soc. 20 (2007), 219--265.
  35. U. Haagerup, Orthogonal maximal abelian ∗*-subalgebras of the n×nn\times n matrices and cyclic nn-roots, in ``Operator algebras and quantum field theory'', International Press (1997), 296--322.
  36. L.M. Higginbotham and C.T. Worley, On some quaternionic Hadamard matrices of small order, 2112.09613 .
  37. S. Hopkins, Sequential deformations of Hadamard matrices and commuting squares, PhD thesis, Univ. of Tennessee (2022).
  38. S. Hopkins and R. Nicoara, A finiteness result for complex Hadamard matrices in maximal abelian self-adjoint subalgebras of Mn(C)M_n(\mathbb C), preprint.
  39. S. Hopkins and R. Nicoara, On the structure of the moduli space of commuting squares around the Fourier spin model, preprint.
  40. M. Izumi, S. Morrison and D. Penneys, Quotients of A2∗T2A_2*T_2, Canad. J. Math. 68 (2016), 999--1022.
  41. F. Jaeger, M. Matsumoto and K. Nomura, Bose--Mesner algebras related to type II matrices and spin models, J. Algebraic Combin. 8 (1998), 39--72.
  42. V.F.R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1--25.
  43. V.F.R. Jones, Planar algebras I, New Zealand J. Math. 52 (2021), 1--107.
  44. V.F.R. Jones, The annular structure of subfactors, Monogr. Enseign. Math. 38 (2001), 401--463.
  45. V.F.R. Jones and V.S. Sunder, Introduction to subfactors, London Math. Soc. Lect. Notes 234 , Cambridge Univ. Press (1997).
  46. B.R. Karlsson, H2H_2-reducible complex Hadamard matrices of order 6, Linear Algebra Appl. 434 (2011), 239--246.
  47. B.R. Karlsson, Three-parameter complex Hadamard matrices of order 6, Linear Algebra Appl. 434 (2011), 247--258.
  48. W. Liu, A noncommutative de Finetti theorem for boolean independence, J. Funct. Anal. 269 (2015), 1950--1994.
  49. R. Mathon, Symmetric conference matrices of order pq2+1pq^2+1, Canad. J. Math. 30 (1978), 321--331.
  50. B.D. McKay, Hadamard matrices, data library, https://users.cecs.anu.edu.au/~bdm/data/hadamard.html.link
  51. D. McNulty and S. Weigert, Isolated Hadamard matrices from mutually unbiased product bases, J. Math. Phys. 53 (2012), 032202.
  52. M. Montgomery, Angle operators of commuting square subfactors, PhD thesis, Vanderbilt Univ. (2022).
  53. M. Montgomery, Colored planar algebras for commuting squares and applications to Hadamard subfactors, J. Funct. Anal. 285 (2023), 110010.
  54. M. Montgomery, The spectrum of spin model angle operators, 2110.06720 .
  55. A. Nica and R. Speicher, Lectures on the combinatorics of free probability, London Math. Soc. Lect. Notes 335 , Cambridge Univ. Press (2006).
  56. R. Nicoara, Subfactors and Hadamard matrices, J. Operator Theory 64 (2010), 453--468.
  57. R. Nicoara and J. White, Twisted commuting squares, Internat. J. Math. 24 (2013), 1350036.
  58. R. Nicoara and J. White, The defect of a group-type commuting square, Rev. Roumaine Math. Pures Appl. 59 (2014), 245--254.
  59. R. Nicoara and J. White, Analytic deformations of group commuting squares and complex Hadamard matrices, J. Funct. Anal. 272 (2017), 3486--3505.
  60. R. Nicoara and J. White, Second order deformations of group commuting squares and Hadamard matrices, Proc. Amer. Math. Soc. 148 (2020), 3967--3974.
  61. R. Nicoara and C. Worley, A finiteness result for circulant core complex Hadamard matrices, Linear Algebra Appl. 571 (2019), 143--153.
  62. K. Nomura, An algebra associated with a spin model, J. Algebraic Combin. 6 (1997), 53--58.
  63. A. Ocneanu, Quantized groups, string algebras and Galois theory for algebras, London Math. Soc. Lect. Notes 136 (1988), 119--172.
  64. K. Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc. 350 (1998), 1205--1223.
  65. R. Paley, On orthogonal matrices, J. Math. Phys. 12 (1933), 311--320.
  66. M. Petrescu, Existence of continuous families of complex Hadamard matrices of certain prime dimensions and related results, PhD thesis, UCLA (1997).
  67. S. Popa, Orthogonal pairs of ∗*-subalgebras in finite von Neumann algebras, J. Operator Theory 9 (1983), 253--268.
  68. T. Sano and Y. Watatani, Angles between two subfactors, J. Operator Theory 32 (1994), 209--241.
  69. G. Schmidt, Classification results of Hadamard matrices, MS thesis, Univ. of Tennessee (2017).
  70. H.-J. Schneider, Some remarks on exact sequences of quantum groups, Comm. Algebra 21 (1993), 3337--3357.
  71. F. Szöllősi, Parametrizing complex Hadamard matrices, European J. Combin. 29 (2008), 1219--1234.
  72. F. Szöllősi, Exotic complex Hadamard matrices and their equivalence, Cryptogr. Commun. 2 (2010), 187--198.
  73. F. Szöllősi, Complex Hadamard matrices of order 6: a four-parameter family, J. Lond. Math. Soc. 85 (2012), 616--632.
  74. W. Tadej and K. Życzkowski, A concise guide to complex Hadamard matrices, Open Syst. Inf. Dyn. 13 (2006), 133--177.
  75. W. Tadej and K. Życzkowski, Defect of a unitary matrix, Linear Algebra Appl. 429 (2008), 447--481.
  76. T. Tao, Fuglede's conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), 251--258.
  77. P. Tarrago and J. Wahl, Free wreath product quantum groups and standard invariants of subfactors, Adv. Math. 331 (2018), 1--57.
  78. H.N.V. Temperley and E.H. Lieb, Relations between the ``percolation'' and ``colouring'' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ``percolation'' problem, Proc. Roy. Soc. London Ser. A 322 (1971), 251--280.
  79. S. Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), 671--692.
  80. S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195--211.
  81. H. Wenzl, On sequences of projections, C. R. Math. Rep. Acad. Sci. Canada 9 (1987), 5--9.
  82. J. White, Isolation and deformation results for commuting squares of finite dimensional ∗*-algebras, PhD thesis, Univ. of Tennessee (2014).
  83. C. Worley, Construction and classification results for commuting squares of finite dimensional ∗*-algebras, PhD thesis, Univ. of Tennessee (2017).
  84. S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613--665.
  85. S.L. Woronowicz, Tannaka--Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93 (1988), 35--76.

Version history

  1. v1 · 58 pagesSubmitted by Teo BanicaInitial depositCurrentNov 13, 2026