Enumeration Techniques on Cyclic Schur Rings

A. Misseldine

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 4, pp. 276-289

Résumé

This paper is based on the results of the 2020 Ural Workshop on Group Theory and Combinatorics. Any Schur ring is uniquely determined by a partition of the elements of the group. In this paper we present a general technique for enumerating Schur rings over cyclic groups using traditional Schur rings. We also survey recent efforts to enumerate Schur rings over cyclic groups of specific orders.

Détail

Keywords: Schur ring; cyclic group; association scheme; lattice of subgroups.

Bibliographie

[1] Ashrafi A.R. and Koorepazan-Moftakhar F., “Towards the classification of finite simple groups with exactly three or four supercharacter theories”, Asian-Eur. J. Math., 11:5 (2018), 1850096, 21 pp. | DOI | Zbl

[2] Ashrafi Ali Reza and Koorepazan-Moftakhar Fatemeh, “Denombrement des theories de supercaracteres d'un groupe fini”, C. R. Math. Acad. Sci. Paris, 357:4 (2019), 323–326 | DOI | Zbl

[3] Bastian Nicholas, Brewer Jaden, Humphries Stephen, Misseldine Andrew, and Thompson Cache, “On Schur rings over infinite groups”, Algebras and Representation Theory, 23 (2020), 493–511 | DOI | Zbl

[4] Bosma Wieb, Cannon John, and Playoust Catherine, “The Magma algebra system. I. The user language”, J. Symbolic Comput., 24:3–4 (1997), 235–265 | DOI | Zbl

[5] Burkett Shawn, Lamar Jonathan, Lewis Mark L., and Wynn Casey, “Groups with exactly two supercharacter theories”, Commun. Algebra, 45:3 (2017), 977–982 | DOI | Zbl

[6] Calugareanu Grigore, “The total number of subgroups of a finite Abelian group”, Sci. Math. Jpn., 60:1 (2004), 157–167 | Zbl

[7] Evdokimov S.A. and Ponomarenko I.N., “On a family of Schur rings over a finite cyclic group”, St. Petersbg. Math. J., 13:3 (2002), 441–451 | Zbl

[8] Evdokimov S.A. and Ponomarenko I.N., “Characterization of cyclotomic schemes and normal Schur rings over a cyclic group”, St. Petersbg. Math. J., 14:2 (2003), 11–55 | Zbl

[9] Evdokimov S.A. and Ponomarenko I.N., “Schurity of S-rings over a cyclic group and generalized wreath product of permutation groups”, St. Petersbg. Math. J., 24:3 (2013), 431–460 | DOI | Zbl

[10] Evdokimov Sergei, Kovács István, and Ponomarenko Ilya, “On schurity of finite Abelian groups”, Commun. Algebra, 44:1 (2016), 101–117 | DOI | Zbl

[11] Goursat E., “Sur les substitutions orthogonales et les divisions regulieres de l'espace”, Ann. Sci. Ec. Norm. Super.(3), 6 (1889), 9–102 | DOI

[12] Hendrikson Anders O. F., “Supercharacter theory constructions corresponding to Schur ring products”, Commun. Algebra, 40 (2012), 4420–4438 | DOI

[13] Humphries Stephen P. and Wagner David R., “Central Schur rings over the projective special linear groups”, Commun. Algebra, 45:12 (2017), 5325–5337 | DOI | Zbl

[14] Keller Joseph, Misseldine Andrew, and Sullivan Max, “Counting Schur rings over semi-prime order”, Preprint, 2020, 18 pp., arXiv: math.2005.06373

[15] Kerby Brent, “Rational Schur rings over Abelian groups”, Theses and Dissertations, 2008, 1491, 90 pp. URL: https://scholarsarchive.byu.edu/etd/1491

[16] Kovács István, “The number of indecomposable Schur rings over a cyclic 2-group”, Seminaire Lotharingien de Combinatoire, 5 (2005), B51h, 16 pp.

[17] Lang Alexander, An enumeration of the supercharacter theories of $C_p\times C_2 \times C_2$ for prime $p$, Preprint, arXiv: math.1609.07182

[18] Leung Ka Hin and Ma Siu Lun, “The structure of Schur rings over cyclic groups”, J. Pure and Appl. Algebra, 66 (1990), 287–302 | DOI | Zbl

[19] Leung Ka Hin and Man Shin Hing, “On Schur rings over cyclic groups II”, J. Algebra, 183 (1996), 273–285 | DOI | Zbl

[20] Leung Ka Hin and Man Shin Hing, “On Schur rings over cyclic groups”, Israel J. Math., 106 (1998), 251–267 | DOI | Zbl

[21] Liskovets V. and Pöschel R., “Counting circulant graphs of prime-power order by decomposing into orbit enumeration problems”, Discr. Math., 214 (2000), 173–191 | DOI | Zbl

[22] Misseldine Andrew, “Counting Schur rings over cyclic groups”, J. Algebraic Comb., 51 (2020), 155–169 | DOI | Zbl

[23] Muzychuk Mikhail and Ponomarenko Ilia, “Schur rings”, European J. Combin., 30 (2009), 1526–1539 | DOI | Zbl

[24] Muzychuk Mikhail E., “The structure of rational Schur rings over cyclic groups”, European J. Combin., 14 (1993), 479–490 | DOI | Zbl

[25] Muzychuk Mikhail E., “On the structure of basic sets of Schur rings over cyclic groups”, J. Algebra, 169 (1994), 655–678 | DOI | Zbl

[26] Muzychuk Mikhail E., “The structure of Schur rings over cyclic groups of square-free order”, Acta Applicandae Mathematicae, 52 (1998), 163–181 | DOI | Zbl

[27] Oh Ju-Mok, “An explicit formula for the number of subgroups of a finite Abelian p-group up to rank 3”, Commun. Korean Math. Soc., 28:4 (2013), 649–667 | DOI | Zbl

[28] Petrillo Joseph, “Counting subgroups in a direct product of finite cyclic groups”, The College Math. J., 42:3 (2011), 215–222 | DOI | Zbl

[29] Pöschel R., “Investigations of S-rings, especially in the group ring of p-groups”, Math. Message, 60 (1974), 1–27 | Zbl

[30] Schur Issai, “Zur theorie der einfach transitiven permutationsgruppen”, Sitzungsber. Preuss. Akad. Wiss. Phy-Math Klasse, 118 (1933), 309–310, Berlin

[31] Tamaschke Olaf, “On Schur-rings which define a proper character theory on finite groups”, Math. Z., 117 (1970), 340–360 | DOI | Zbl

[32] Tóth László, “Subgroups of finite Abelian groups having rank two via Goursat's lemma”, Tatra Mt. Math. Publ., 59:1 (2014), 93–103 | DOI | Zbl

[33] Wielandt Helmut, “Zur theorie der einfach transitiven permutationsgruppen II”, Math. Z., 52 (1949), 384–393 | DOI | Zbl

[34] Wielandt Helmut, Finite permutation groups, Acad. Press, NY, London, 1964 | Zbl

[35] Ziv-Av Matan, “Enumeration of Schur rings over small groups”, Proc. Computer Algebra in Scientific Computing, 16th International Workshop (CASC 2014), Warsaw, 2014, 491–500 | DOI | Zbl