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On Schur Rings over Infinite Groups

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Abstract

Schur rings are a type of subring of the group ring that is determined by a partition of the group. Past literature has exclusively focused on the finite group case. This paper extends many classic results about Schur rings to infinite groups, including Leung-Man’s classification of Schur rings over finite cyclic groups which is extended to the infinite cyclic group, as well as all torsion-free locally cyclic groups. Schurs rings over free groups and free products are also considered.

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References

  1. Haviv, I., Levy, D.: Dioid partitions of groups, preprint (2018)

  2. Heinze, A.: Applications of Schur Rings in Algebraic Combinatorics: Graphs, Partial Difference Sets and Cyclotomic Schemes. PhD thesis, Universität Oldenburg (2001)

  3. Hendrikson, A.O.F.: Supercharacter theory constructions corresponding to Schur ring products

  4. Hirasaka, M., Muzychuk, Mi.: An elementary Abelian group of rank 4 is a CI-group. J. Combinator. Theory Series A 94, 339–362 (2001)

    Article  MathSciNet  Google Scholar 

  5. Khramtsov, D.G.: Finite groups of automorphisms of free groups. Mat. Zametki 476(3), 386–392 (1985)

    MathSciNet  MATH  Google Scholar 

  6. Klin, M. Kh., Pöschel, R.: The König problem, the isomorphism problem for cyclic graphs and the method of Schur rings. Algebraic Methods Graph Theory 1, 2 (1978)

    MATH  Google Scholar 

  7. Klin, M.Ch., Najmark, N.L., Pöschel, R.: Schur-rings over \(Z_{2}^{m}\). Prepr., Akad. Wiss. DDR, Inst. Math. P-Math-14/81, 31 p (1981)

  8. Ka, H.L., Ma, S.L.: The structure of Schur rings over cyclic groups. J. Pure Appl. Algebra 66, 287–302 (1990)

    Article  MathSciNet  Google Scholar 

  9. Ka, H.L., Man, S.H.: On Schur rings over cyclic groups II. J. Algebra 183, 273–285 (1996)

    Article  MathSciNet  Google Scholar 

  10. Ka, H.L., Man, S.H.: On Schur rings over cyclic groups. Israel J. Math. 106, 251–267 (1998)

    Article  MathSciNet  Google Scholar 

  11. Lopez, R.O.: An approximation to proof of the circulant Hadamard conjecture, preprint (2018)

  12. Lopez, R.O.: Schur ring over group \(\mathbb {Z}_{2}^{n}\), circulant S −sets invariant by decimation and Hadamard matrices, preprint (2018)

  13. Lyndon, R.C., Schupp, P.E.: Combinatorial group theory (1977)

  14. Misseldine, A.: Algebraic and Combinatorial Properties of Schur Rings over Cyclic Groups. PhD thesis, Brigham Young University (2014)

  15. Misseldine, A.: Primitive idempotents of Schur rings. Algebras Represent. Theory 17(5), 1615–1634 (2014)

    Article  MathSciNet  Google Scholar 

  16. Muzychuk, M., Ponomarenko, I.: Schur rings. Europ. J. Combinator. 30, 1526–1539 (2009)

    Article  MathSciNet  Google Scholar 

  17. Muzychuk, M.E.: On the structure of basic sets of Schur rings over cyclic groups. J. Algebra 169, 655–678 (1994)

    Article  MathSciNet  Google Scholar 

  18. Pöschel, R.: Untersuchungen von S-Ringen, insbesondere im Gruppenring von p-Gruppen. Math. Nachr. 60, 1–27 (1974)

    Article  MathSciNet  Google Scholar 

  19. Saloff-Coste, L.: Probability on groups:random walks and invariant diffusions. Not. Am. Math. Soc. 48, 968–977 (2001)

    MathSciNet  MATH  Google Scholar 

  20. Schur, I.: Zur theorie der einfach transitiven permutationsgruppen. Sitzungsber. Preuss. Akad. Wiss. Phy-Math Klasse. Berlin 118, 309–310 (1933)

    MATH  Google Scholar 

  21. Scott, W.R.: Group Theory. Dover Publications, Inc., New York (1987)

    Google Scholar 

  22. Shiu, W.-C.: Schur rings over dihedral groups. Chin. J. Math. 18(3), 209–223 (1990)

    MathSciNet  MATH  Google Scholar 

  23. Tamaschke, O.: On the theory of Schur-rings. Ann. Mat. Pura Appl. (4) 81, 1–43 (1969)

    Article  MathSciNet  Google Scholar 

  24. Tamaschke, O.: On Schur-rings which define a proper character theory on finite groups. Math. Z. 117, 340–360 (1970)

    Article  MathSciNet  Google Scholar 

  25. Wielandt, H.: Zur theorie der einfach transitiven permutationsgruppen II (German). Math. Z. 52, 384–393 (1949)

    Article  MathSciNet  Google Scholar 

  26. Wielandt, H.: Finite Permutation Groups. Academic Press, New York (1964)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Southern Utah University, Cedar City, UT, USA

    Nicholas Bastian, Jaden Brewer & Andrew Misseldine

  2. Brigham Young University, Provo, UT, USA

    Stephen Humphries & Cache Thompson

Authors
  1. Nicholas Bastian
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  2. Jaden Brewer
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  3. Stephen Humphries
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  4. Andrew Misseldine
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  5. Cache Thompson
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Correspondence to Andrew Misseldine.

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Presented by: Anne Schilling

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Bastian, N., Brewer, J., Humphries, S. et al. On Schur Rings over Infinite Groups. Algebr Represent Theor 23, 493–511 (2020). https://doi.org/10.1007/s10468-019-09859-7

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