Previous |  Up |  Next


Title: Loop characters (English)
Author: Johnson, Kenneth W.
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 41
Issue: 2
Year: 2000
Pages: 271-281
Category: math
Summary: A survey of the basic results of loop characters is given on the lines of the treatment of the author and J.D.H. Smith for characters of quasigroups, including some recent deveploments. One of the successes of the theory has been its suggestive influence on the theory of association schemes, group representations and the theory of the group determinant, and selected results arising are described. A section is devoted to an explanation of how the tool of loop characters has not yet been as startlingly successful as that of the early theory of group characters. This may be because in the loop case more is needed than characters and some suggestions are put forward in this direction. (English)
Keyword: loop
Keyword: character
Keyword: association scheme
MSC: 05E30
MSC: 19A22
MSC: 20C99
MSC: 20N05
idZBL: Zbl 1038.20048
idMR: MR1780871
Date available: 2009-01-08T19:01:11Z
Last updated: 2012-04-30
Stable URL:
Reference: [1] Bannai E., Ito T.: Algebraic Combinatorics I..Benjamin-Cummings Lecture Notes Series No. 58, Menlo Park, California, 1994. Zbl 0685.05030, MR 0882540
Reference: [2] Bannai E., Song S.: The character tables of Paige's simple Moufang loops and their relationship to the character tables of PSL$(2,q)$.Proc. London Math. Soc. (3) 58 (1989), 209-236. Zbl 0682.20050, MR 0977475
Reference: [3] Bannai E., Kawanaka N., Song S.: The character table of the Hecke algebra ${\Cal H}({GL}_{2n}(F_ q),{Sp}_ {2n} (F_ q))$.J. Algebra 129 (1990), 320-366. MR 1040942
Reference: [4] Bannai E., Hao S., Song S.: Character tables of the association schemes of finite orthogonal groups acting on the nonisotropic points.J. Combin. Theory Ser. A 54 (1990), 164-200. Zbl 0762.20005, MR 1059994
Reference: [5] Brauer R.: Representations of finite Lectures in Modern Mathematics, Vol. 1, T.L. Saaty (ed.), Wiley, 1963, pp.133-175. Zbl 0333.20008, MR 0178056
Reference: [6] Cameron P.J., Kiyota M.: Sharp characters of finite groups.J. Algebra 115 (1988), 125-143. Zbl 0651.20010, MR 0937604
Reference: [7] Godsil C.D.: Algebraic Combinatorics.Chapman & Hall, New York, 1993. Zbl 0814.05075, MR 1220704
Reference: [8] Formanek E., Sibley D.: The group determinant determines the group.Proc. Amer. Math. Soc 112 (1991), 649-656. Zbl 0742.20008, MR 1062831
Reference: [9] Frobenius G.: Über Gruppencharaktere.Sitzungsber. Preuss. Akad. Wiss. Berlin (1896), 985-1021. (Gesammelte Abhandlungen, (Springer-Verlag 1968), pp.1-37).
Reference: [10] Johnson K.W.: Latin square Algebraic, Extremal and Metric Combinatorics 1986, London Math. Soc. Lecture Notes 131, 1988, pp.146-154. Zbl 0761.05019, MR 1052664
Reference: [11] Johnson K.W.: Some historical aspects of the development of group representation theory and its extension to quasigroups.Universal Algebra and Quasigroup Theory, A. Romanowska, J.D.H. Smith (eds.) Heldermann Verlag, Berlin, 1992, pp.101-117. MR 1191229
Reference: [12] Johnson K.W.: Latin square determinants II..Discrete Math. 105 (1992), 111-130. Zbl 0761.05019, MR 1180197
Reference: [13] Johnson K.W.: Sharp characters of quasigroups.European J. Combin. 14 (1993), 103-112. Zbl 0773.20030, MR 1206615
Reference: [14] Johnson K.W.: The Dedekind-Frobenius group determinant, new life in an old method.Proceedings of the Groups 97 Conference, Bath, England 1997; London Math. Soc. Lecture Notes 261, 1999, pp.417-428. MR 1676638
Reference: [15] Johnson K.W., Ford D.: Determinants of latin squares of order $8$.Experimental Math. 5 (1996), 317-325. Zbl 0876.05017, MR 1437221
Reference: [16] Johnson K.W., Mattarei S., Sehgal S.K.: Weak Cayley appear in J. London Math. Soc. Zbl 0962.20003
Reference: [17] Johnson K.W., Poimenidou E.: Generalised classes in groups and association schemes, duals of results on characters and sharpness.European J. Combin. 20 (1999), 1-6. Zbl 0916.05075, MR 1669612
Reference: [18] Johnson K.W., Smith J.D.H.: Characters of finite quasigroups.European J. Combin. 5 (1984), 43-50. Zbl 0537.20042, MR 0746044
Reference: [19] Johnson K.W., Smith J.D.H.: Characters of finite quasigroups II: induced characters.European J. Combin. 7 (1986), 131-138. Zbl 0599.20110, MR 0856325
Reference: [20] Johnson K.W., Smith J.D.H.: Characters of finite quasigroups III, Quotients and fusion.European J. Combin. 10 (1989), 47-56. Zbl 0667.20053, MR 0977179
Reference: [21] Johnson K.W., Smith J.D.H.: Characters of finite quasigroups IV: products and superschemes.European J. Combin. 10 (1989), 257-263. Zbl 0669.20053, MR 1029172
Reference: [22] Johnson K.W., Smith J.D.H.: Characters of finite quasigroups V: linear characters.European J. Combin. 10 (1989), 449-456. Zbl 0679.20059, MR 1014553
Reference: [23] Johnson K.W., Smith J.D.H.: A note on the induction of characters in association schemes.European J. Combin. 7 (1986), 139-140. MR 0856326
Reference: [24] Johnson K.W., Song S-Y., Smith J.D.H.: Characters of finite quasigroups VI: critical examples.European J. Combin. 11 (1990), 267-275. Zbl 0704.20056, MR 1059557
Reference: [25] Mackey G.W.: Harmonic analysis as an exploitation of symmetry - a historical survey.Bull. Amer. Math. Soc. 3 (1980), 543-698. MR 0571370
Reference: [26] Smith J.D.H.: Combinatorial characters of Coding Theory and Design Theory, Part I: Coding Theory, (D. Ray-Chaudhuri, ed.) Springer, New York, 1990. Zbl 0708.20021, MR 1047879
Reference: [27] Smith J.D.H.: Quasigroup actions: Markov chains, pseudoinverses, and linear appear in South-East Asian Bulletin of Mathematics.
Reference: [28] Smith J.D.H.: Quasigroup representation theory. Universal Algebra and Quasigroup Theory.A. Romanowska, J.D.H. Smith (eds.), Heldermann, Berlin, 1992, pp.195-207. MR 1191234
Reference: [29] Song S-Y.: Fusion relations in products of association schemes.preprint. MR 1939083


Files Size Format View
CommentatMathUnivCarolRetro_41-2000-2_7.pdf 219.3Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo