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A survey of semisimple algebras in algebraic combinatorics

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Abstract

This is a survey of semisimple algebras of current interest in algebraic combinatorics, with a focus on questions which we feel will be new and interesting to experts in group algebras, integral representation theory, and computational algebra. The algebras arise primarily in two families: coherent algebras and subconstituent (aka. Terwilliger) algebras. Coherent algebras are subalgebras of full matrix algebras having a basis of 01-matrices satisfying the conditions that it be transpose-closed, sum to the all 1’s matrix, and contain a subset \(\Delta \) that sums to the identity matrix. The special case when \(\Delta \) is a singleton is the important case of an adjacency algebra of a finite association scheme. A Terwilliger algebra is a semisimple extension of a coherent algebra by a set of diagonal 01-matrices determined canonically from its basis elements and a choice of row. We will survey the current state of knowledge of the complex, real, rational, modular, and integral representation theory of these semisimple algebras, indicate their connections with other areas of mathematics, and present several open questions.

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References

  1. Z. Arad and E. Fisman, On table algebras, C-algebras, and applications to finite group theory, Comm. Algebra, 19, (1991), 2955-3009.

    Article  MathSciNet  Google Scholar 

  2. Z. Arad, E. Fisman, and M. Muzychuk, Generalized table algebras, Israel J. Math., 114 (1999), 29-60.

    Article  MathSciNet  Google Scholar 

  3. S. Bang, A. Dubickas, J. Koolen, and V. Moulton, There are only finitely many distance-regular graphs of fixed valency greater than two, Adv. Math., 269 (2015), 1-55.

    Article  MathSciNet  Google Scholar 

  4. S. Bang and M. Hirasaka, Construction of association schemes from difference sets, European J. Combin., 26 (1), (2005), 59–74.

    Article  MathSciNet  Google Scholar 

  5. E. Bannai, Combinatorics as Pure Mathematics - Aims of Algebraic Combinatorics, Sugaku, 62 (4), (2010), 433-452. (Japanese, English version at http://math.sjtu.edu.cn/conference/Bannai/2012/data/journal.pdf)

  6. E. Bannai, Subschemes of some association schemes, J. Algebra, 144 (1), (1991), 167–188.

    Article  MathSciNet  Google Scholar 

  7. E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin Cummings, Menlo Park, CA, 1984.

    MATH  Google Scholar 

  8. E. Bannai and S.-Y. Song, Character tables of fission schemes and fusion schemes, European J. Combin., 14 (5), (1993), 385–396.

    Article  MathSciNet  Google Scholar 

  9. H. Blau, Table algebras, European J. Combin., 30, (2009), 1426-1455.

    Article  MathSciNet  Google Scholar 

  10. H. Blau, Quotient structures in \(C\)-algebras, J. Algebra, 175 (1995), 24-64; Erratum: 177 (1995), 297-337.

  11. H. Blau and G. Chen, Reality-based algebras, generalized Camina-Frobenius pairs, and the non-existence of degree maps, 40 (4), (2012), 1547-1562.

  12. A. Brouwer, BCN tables, https://www.win.tue.nl/~aeb/drg/drgtables.html

  13. A. Brouwer, A. Cohen, and A. Neumaier, Distance Regular Graphs, Springer-Verlag, Berlin, 1989.

    Book  Google Scholar 

  14. A. Brouwer and C.A. van Eijl, On the p-rank of the adjacency matrices of strongly regular graphs. J. Algebraic Combin., 1 (4), (1992), 329–346.

    Article  MathSciNet  Google Scholar 

  15. G. Birkhoff, Three observations on linear algebra. Univ. Nac. Tucumán. Revista A, 5 (1946). 147–151. (Spanish)

  16. C. Bushnell and I. Reiner, Zeta functions of arithmetic orders and Solomon’s conjectures, Math. Z., 173, (1980), 135-161.

    Article  MathSciNet  Google Scholar 

  17. A. Coste and T. Gannon, Remarks on Galois in rational conformal field theories, Phys. Lett. B, 323, (1994), 316-321.

    Article  MathSciNet  Google Scholar 

  18. C. Curtis and I. Reiner, Methods of Representation Theory: With applications to groups and orders, Vol. I, Wiley-Interscience, New York, 1991.

    Google Scholar 

  19. C. French, Functors from association schemes, J. Combin. Theory Ser. A, 120 (6), (2013), 1141–1165.

    Article  MathSciNet  Google Scholar 

  20. A. Hanaki, A category of association schemes, J. Combin. Theory Ser. A, 117 (8), (2010), 1207–1217.

    Article  MathSciNet  Google Scholar 

  21. A. Hanaki, Representation theory of finite association schemes, European J. Combin., 30 (6), (2009), 1477-1496.

    Article  MathSciNet  Google Scholar 

  22. A. Hanaki, Locality of a modular adjacency algebra of an association scheme of prime power order, Arch. Math. (Basel), 79 (3), (2002), 167–170.

    Article  MathSciNet  Google Scholar 

  23. A. Hanaki, Semisimplicity of adjacency algebras of association schemes, J. Algebra, 225, (2000), 124-129.

    Article  MathSciNet  Google Scholar 

  24. A. Hanaki and M. Hirasaka, The zeta function of the integral adjacency algebra of association scheme of prime order or rank \(2\), Hokkaido J. Math., 46 (1), (2015), 75-91.

    MATH  Google Scholar 

  25. A. Hanaki and M. Hirasaka, Theory of Hecke algebras to association schemes, SUT J. Math., 38 (1), (2002), 61-66.

    MathSciNet  MATH  Google Scholar 

  26. A. Hanaki, H. Kharaghani, A. Mohammadian, and B. Tayfeh-Rezaie, Classification of skew-Hadamard matrices of order 32 and association schemes of order 31, preprint, available at http://math.ipm.ac.ir/~tayfeh-r/papersandpreprints/SH32AS31r.pdf

  27. A. Hanaki and I. Miyamoto, Classification of Small Association Schemes, http://math.shinshu-u.ac.jp/~hanaki/as/

  28. A. Hanaki and K. Uno, Algebraic structure of association schemes of prime order, J. Algebraic Combin., 23, (2006), 189-195.

    Article  MathSciNet  Google Scholar 

  29. A. Herman, Trivial Schur indices for reality-based algebras with two nonreal elements, preprint, available at arXiv:1905.00445v2 [math.RA]

  30. A. Herman, M. Hirasaka, and S. Oh, Zeta functions for tensor products of locally coprime integral adjacency algebras of association schemes, Comm. Algebra, 45 (11), (2017), 4896-4905.

    Article  MathSciNet  Google Scholar 

  31. A. Herman, M. Muzychuk, and B. Xu, Noncommutative reality-based algebras of rank \(6\), Comm. Algebra, 46 (1), (2018), 90-113.

    Article  MathSciNet  Google Scholar 

  32. A. Herman, M. Muzychuk, and B. Xu, The recognition problem for table algebras and reality-based algebras, J. Algebra, 479 (2017), 173-191.

    Article  MathSciNet  Google Scholar 

  33. A. Herman and A. Rahnamai Barghi, Schur indices of association schemes, J. Pure Appl. Algebra, (2011),

  34. A. Herman and G. Singh, Central torsion units of integral reality-based algebras with a positive degree map, Intl. Elec. J. Algebra, 21, (2017), 121-126.

    Article  MathSciNet  Google Scholar 

  35. D.G. Higman, Coherent algebras, Linear Algebra Appl., 93 (1987), 209-239.

    Article  MathSciNet  Google Scholar 

  36. P. Etinghof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Mathematical Surveys and Monographs, Vol. 205, American Mathematical Society, Providence, RI, 2015.

    Book  Google Scholar 

  37. Y. Hironaka, Zeta functions of integral group rings of metacyclic groups, Tsukuba J. Math., 5 (2), (1981), 267-283; Corrections Tskuba J. Math., 9 (2), (1985), 373-374.

  38. M. Klin and M. Ziv-Av, A non-Schurian coherent configuration on 14 points exists, Des. Codes Cryptogr., 84 (1-2), (2017), 203–221.

    Article  MathSciNet  Google Scholar 

  39. T. Komatsu, Tamely ramified Eisenstein fields with prime power discriminants, Kyushu J. Math., 62 (1), (2008), 1–13.

    Article  MathSciNet  Google Scholar 

  40. M. Linckelmann, The Block Theory of Finite Group Algebras, London Mathematical Society Student Texts, Vol. 91-92, Cambridge University Press, Cambridge, UK, 2018.

    Google Scholar 

  41. G. Lusztig, Leading coefficients of character values of Hecke algebras, in: Proc. Sympos. Pure Math., Vol. 47, American Mathematical Society, Providence, RI, 1987, pp. 235-273, Part II.

  42. L. Solomon, Zeta functions and integral representation theory, Adv. Math., 26, (1977), 306-326.

    Article  MathSciNet  Google Scholar 

  43. W. Martin and H. Tanaka, Commutative association schemes, European J. Combin., 30 (6), (2009), 1497-1525.

    Article  MathSciNet  Google Scholar 

  44. M. Muzychuk, V-rings of permutation groups with invariant metric. Ph.D. thesis, Kiev State University, 1987.

  45. I. Reiner, Zeta functions of integral representations, Comm. Algebra, 8 (10), (1980), 911-925.

    Article  MathSciNet  Google Scholar 

  46. Y. Takegahara, Zeta functions of integral group rings of abelian \((p,p)\)-groups, Comm. Algebra, 15 (12), (1987), 2565-2615.

  47. P. Terwilliger, The subconstituent algebra of an association scheme I, J. Alg. Combin., 1 (1992), 363-388.

    Article  MathSciNet  Google Scholar 

  48. J. Vidali, Using symbolic computation to prove nonexistence of distance-regular graphs, Electron. J. Combin., 25, (4), (2018), Paper 4.21, 10 pp.

  49. P. Terwilliger, The subconstituent algebra of an association scheme III, J. Alg. Combin., 2 (1993), 177-210.

    Article  MathSciNet  Google Scholar 

  50. M. Tomiyama and N. Yamazaki, The subconstituent algebra of a strongly regular graph, Kyushu J. Math., 48 (2), (1994), 323-334.

    Article  MathSciNet  Google Scholar 

  51. B. Xu, Wedge-direct products of table algebras with applications to association schemes I, 47 (8), Comm. Algebra, (2019), 3309-3328.

  52. B. Xu, Table algebras with central fusions, Comm. Algebra, 46 (3), (2018), 1018-1031.

    Article  MathSciNet  Google Scholar 

  53. B. Xu, On maximal actions and \(\omega \)-maximal actions of finite hypergroups, J. Algebraic Combin., 27, (2008), 127-141.

    Article  MathSciNet  Google Scholar 

  54. P.-H. Zieschang, Theory of Association Schemes, Springer-Verlag, Berlin, 2005.

    MATH  Google Scholar 

  55. P.-H. Zieschang, Algebraic Structure of Association Schemes, Lecture Notes in Mathematics No. 1628, Springer-Verlag, Berlin, 1996.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Regina, Regina, SK, S4A 0A2, Canada

    Allen Herman

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  1. Allen Herman
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Correspondence to Allen Herman.

Additional information

Communicated by Gadadhar Misra.

The author’s work has been supported by NSERC Discovery Grant.

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Herman, A. A survey of semisimple algebras in algebraic combinatorics. Indian J Pure Appl Math 52, 631–642 (2021). https://doi.org/10.1007/s13226-021-00181-x

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