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Augmented quasigroups and character algebras. (English) Zbl 1471.20047

The paper addresses generalizations of the character theory of abelian groups. It is well-known that there is a duality between an abelian group \(G\) and its character group \(G^*=\mathrm{Hom}(G,\mathbb{C})\) which is exploited in harmonic analysis. For a non-abelian group there have been diverse attempts to generalize this relationship but it has proved difficult to obtain a satisfactory theory.
The paper provides an intricate exposition of the algebra, combinatorics and category theory behind the discussion of duality.
In the case of groups, duality can be approached by considering the association scheme which arises from the group acting on itself by left and right multiplication. A more general point of view is obtained by considering a quasigroup \(Q\) (where neither associativity nor identity element may be present) but where an association scheme can be produced with a character theory which in the case where \(Q\) is a group coincides with the usual characters. Not all association schemes can arise in the more general context.
The paper states three main goals: firstly to show every association scheme lifts to a quasigroup, secondly to find an analogue for the character group in the case of an arbitrary finite group and thirdly to “bring order into the abundance of diferent algebras that appear in algebraic combinatorics”. Two such algebras are character algebras and fusion algebras.
Salient results are (a) that every association scheme may be obtained by collapsing a quasigroup multiplication, (b) that for each finite group \(G\) character quasigroups can be constructed which encode the multiplicative structure of the characters of \(G\) (in the case where \(G\) is abelian \(G^*\) is its unique character group) and (c) a fusion algebra and a character algebra each give rise to an augmented quasigroup.
The following concepts appear, in the context of compact closed categories: augmented comagmas, augmented magmas and augmented quasigroups.
Augmented magmas are motivated by the example of the group algebra of a finite group. It is shown that magmas and their set-valued analogues hypermagmas appear as augmented magmas in the compact closed category Rel of relations on sets.
The theory is illustrated using the category of finite dimensional vector spaces over a finite field as well as Rel.
Other structures which appear are weighted magmas and hypermagmas (here the binary relation on a set \(A\) takes values in the power set of \(A\)). It is shown that association schemes are augmented magmas.
The example of the character quasigroups associated to the symmetric group \(S_3\) is given: these are the cyclic group \(C_6\) and intercalates of its multiplication table.
An appendix explains ideas going back to Baer expressing quasigroups as “quotients” of groups.
The paper is a valuable contribution in that it brings in a subtle blend of results from a wide range of contexts.

MSC:

20N05 Loops, quasigroups
05E30 Association schemes, strongly regular graphs
05E10 Combinatorial aspects of representation theory
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
20C15 Ordinary representations and characters
20N20 Hypergroups
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[1] Baer, R., Nets and groups I, Trans. Am. Math. Soc., 46, 110-141 (1939) · JFM 65.0819.02
[2] Bannai, E., Association schemes and fusion algebras (an introduction), J. Algebraic Comb., 2, 327-344 (1993) · Zbl 0790.05098
[3] Bannai, E.; Ito, T., Algebraic Combinatorics (1984), Benjamin-Cummings: Benjamin-Cummings Menlo Park, CA · Zbl 0555.05019
[4] Blau, H. I., Table algebras, Eur. J. Comb., 30, 1426-1455 (2009) · Zbl 1229.05291
[5] Comer, S. D., Hyperstructures associated with character algebras and color schemes, (New Frontiers in Hyperstructures. New Frontiers in Hyperstructures, Molise, 1995 (1996), Hadronic Press: Hadronic Press Palm Harbor, FL), 49-66 · Zbl 0887.20039
[6] Day, B. J., Note on compact closed categories, J. Aust. Math. Soc., 24, 309-311 (1977) · Zbl 0397.18008
[7] Frobenius, G., Über Gruppenharaktere, (Ges Abh. III (1896), Sitzungsber. Preuss. Akad. Wiss.: Sitzungsber. Preuss. Akad. Wiss. Berlin), 985-1021, Ges Abh. III, 1-37
[8] Hilton, A. J.W., Outlines of Latin squares, (Colbourn, C. J.; Mathon, R. A., Combinatorial Design Theory, North-Holland Math. Stud., vol. 149. Combinatorial Design Theory, North-Holland Math. Stud., vol. 149, Ann. Discrete Math., vol. 34 (1987), North-Holland: North-Holland Amsterdam), 225-241 · Zbl 0631.05009
[9] Hilton, A. J.W.; Wojciechowski, J., Weighted quasigroups, (Walker, K., Surveys in Combinatorics. Surveys in Combinatorics, 1993, Keele. Surveys in Combinatorics. Surveys in Combinatorics, 1993, Keele, London Math. Soc. Lecture Note Ser., vol. 187 (1993), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 137-171 · Zbl 0788.05008
[10] Hirasaka, M.; Muzychuk, M.; Zieschang, P.-H., A generalization of Sylow’s theorems on finite groups to association schemes, Math. Z., 241, 665-672 (2002) · Zbl 1010.05082
[11] Hoheisel, G., Über Charaktere, Monatshefte Math. Phys., 48, 448-456 (1939) · JFM 65.1121.01
[12] Im, B.; Ryu, J.-Y.; Smith, J. D.H., Sharply transitive sets in quasigroup actions, J. Algebraic Comb., 33, 81-93 (2011) · Zbl 1214.20062
[13] Jay, C. B., Languages for monoidal categories, J. Pure Appl. Algebra, 59, 61-85 (1989) · Zbl 0693.18003
[14] Jipsen, P.; Kinyon, M., Nonassociative right hoops, Algebra Univers., 80, Article 47 pp. (2019) · Zbl 07123189
[15] Johnson, K. W.; Smith, J. D.H., Characters of finite quasigroups, Eur. J. Comb., 5, 43-50 (1984) · Zbl 0537.20042
[16] Johnson, K. W.; Smith, J. D.H.; Song, S. Y., Characters of finite quasigroup VI: critical examples and doubletons, Eur. J. Comb., 11, 267-275 (1990) · Zbl 0704.20056
[17] Johnstone, P. T., Stone Spaces (1982), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0499.54001
[18] Kawada, Y., Über den Dualitätssatz der Charaktere nichtkommutativer Gruppen, Proc. Phys. Math. Soc. Jpn., 24, 97-109 (1942) · Zbl 0063.03172
[19] Kelly, G. M.; Laplaza, M. L., Coherence for compact closed categories, J. Pure Appl. Algebra, 19, 193-213 (1980) · Zbl 0447.18005
[20] Kinyon, M. K.; Smith, J. D.H.; Vojtěchovský, P., Sylow theory for quasigroups II: sectional action, J. Comb. Des., 25, 159-184 (2017) · Zbl 1373.20087
[21] Lee, H.-Y.; Im, B.; Smith, J. D.H., Stochastic tensors and approximate symmetry, Discrete Math., 340, 1335-1350 (2017) · Zbl 1369.05027
[22] Ljubič, Ju. I., Algebraic methods in evolutionary genetics, Biom. J., 20, 511-529 (1978) · Zbl 0401.92010
[23] Majid, S., A Quantum Groups Primer (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1037.17014
[24] Marty, F., Sur les groupes et hypergroupes attachés à une fraction rationnelle, Ann. Sci. Éc. Norm. Supér., 53, 83-123 (1936) · JFM 62.0666.03
[25] Reed, M. L., Algebraic structure of genetic inheritance, Bull. Am. Math. Soc., 34, 107-130 (1997) · Zbl 0876.17040
[26] Romanowska, A. B.; Smith, J. D.H., Diagrammatic duality, (Topology, Algebra and Categories in Logic. Topology, Algebra and Categories in Logic, Prague (2017)) · Zbl 1459.03105
[27] Serre, J.-P., Linear Representations of Finite Groups (1977), Springer: Springer New York, NY · Zbl 0355.20006
[28] Smith, J. D.H., An Introduction to Quasigroups and Their Representations (2007), Chapman and Hall/CRC: Chapman and Hall/CRC Boca Raton, FL · Zbl 1122.20035
[29] Smith, J. D.H., Sylow theory for quasigroups, J. Comb. Des., 23, 115-133 (2015) · Zbl 1331.20076
[30] Smith, J. D.H.; Romanowska, A. B., Post-Modern Algebra (1999), Wiley: Wiley New York, NY · Zbl 0946.00001
[31] Snaith, V. P., Explicit Brauer Induction (1994), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0991.20005
[32] Tahan, M.a.; Davvaz, B., Algebraic hyperstructures associated to biological inheritance, Math. Biosci., 285, 112-118 (2017) · Zbl 1361.92050
[33] Wall, H. S., Hypergroups, Am. J. Math., 59, 77-98 (1937) · JFM 63.0063.01
[34] de Werra, D., A few remarks on chromatic scheduling, (Roy, B., Combinatorial Programming: Methods and Applications (1975), Reidel: Reidel Dordrecht), 337-342 · Zbl 0314.05104
[35] Wörz-Buzekros, A., Algebras in Genetics (1980), Springer: Springer Berlin · Zbl 0431.92017
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