Abstract
Table algebras all of whose nonidentity basis elements are involutions (in the sense of Zieschang), which serve as a counterpoint to the generic Hecke algebras parametrized by Coxeter groups, are classified. If two-generated, they are the family H n (for all n ≥ 3), which for suitable n arise from schemes defined by affine planes of order n − 1. Otherwise, the basis involutions correspond to the points of a finite projective space whose incidence geometry determines the algebra multiplication. This generalizes to table algebras a previous result of van Dam for association schemes. An algebraic characterization is also given.
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References
Arad Z., Fisman E., Muzychuk M.: Generalized table algebras. Israel J. Math 114, 29–60 (1999)
Blau H.I.: Table algebras. European J. Combin. 30, 1426–1455 (2009)
Blau H.I., Chen G.: Table bases as unions of proper closed subsets. Algebr. Represent. Theory 17, 1527–1552 (2014)
Blau H.I., Zieschang P.-H.: Sylow theory for table algebras, fusion rule algebras, and hypergroups. J. Algebra 273, 551–570 (2004)
van Dam E.R.: A characterization of association schemes from affine spaces. Des. Codes Cryptogr. 21, 83–86 (2000)
van Dam E.R., Muzychuk M.: Some implications on amorphic association schemes. J. Combinatorial Theory, Series A 117, 111–127 (2010)
D. Gumber and H. Kalra, The conjugacy class number k(G) - a different perspective, arXiv:1404.4835v1 [math.GR] 18 Apr 2014
Ja. Ju. Gol’fand, A. V. Ivanov and M. H. Klin, Amorphic cellular rings, in: I. A. Faradzev et al (Eds.), Investigations in Algebraic Theory of Combinatorial Objects, Kluwer, Dordrecht, 1994, 167–186
Humphreys J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)
Kalra H., Gumber D.: A note on conjugacy classes of finite groups, Proc. Indian Acad. Sci. (Math Sci.) 124, 31–36 (2014)
Seidenberg A.: Lectures in Projective Geometry. D. Van Nostrand, Princeton (1962)
A. P. Wang, Structure and applications of Coxeter table algebras, Ph.D. Dissertation, Northern Illinois University, in preparation
P.-H. Zieschang, An Algebraic Approach to Association Schemes, Springer Lecture Notes in Math. 1628, Springer-Verlag, Berlin, 1996
Zieschang P.-H.: Theory of Association Schemes. Springer-Verlag, Berlin (2005)
P.-H. Zieschang, Hypergroups, Max-Planck-Institut für Mathematik Preprint Series (97), 2010
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Blau, H.I., Chen, G. All-involution table algebras and finite projective spaces. Arch. Math. 105, 313–322 (2015). https://doi.org/10.1007/s00013-015-0795-9
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DOI: https://doi.org/10.1007/s00013-015-0795-9