Actions of Finite Hypergroups
V.S. Sunder
and N. J. Wildberger
DOI: 10.1023/A:1025107014451
Abstract
This paper is concerned with actions of finite hypergroups on sets. After introducing the definitions in the first section, we use the notion of 
maximal actions 
to characterise those hypergroups which arise from association schemes, introduce the natural sub-class of *-actions of a hypergroup and introduce a geometric condition for the existence of *-actions of a Hermitian hypergroup. Following an insightful suggestion of Eiichi Bannai we obtain an example of the surprising phenomenon of a 3-element hypergroup with infinitely many pairwise inequivalent irreducible *-actions.
maximal actions to characterise those hypergroups which arise from association schemes, introduce the natural sub-class of *-actions of a hypergroup and introduce a geometric condition for the existence of *-actions of a Hermitian hypergroup. Following an insightful suggestion of Eiichi Bannai we obtain an example of the surprising phenomenon of a 3-element hypergroup with infinitely many pairwise inequivalent irreducible *-actions.Pages: 135–151
Keywords: hypergroups; actions; *-actions; association schemes
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References
1. E. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin/Cummings, Menlo Park, 1984.
2. V.F.R. Jones, “Index for subfactors,” Invent. Math. 72 (1983), 1-15.
3. V.F.R. Jones and V.S. Sunder, Introduction to Subfactors, Cambridge University Press, 1997.
4. Y. Kawada, “ \ddot Uber den Dualitatssatz der Charaktere nichtcommutative Gruppen,” Proc. Phys. Math. Soc. Japan 24(3) (1942), 97-109.
5. N. Obata and N.J. Wildberger, “Generalized hypergroups and orthogonal polynomials,” Nagoya Math. J. 142 (1996), 67-93.
6. V.S. Sunder, “II1 factors, their bimodules and hypergroups,” Trans. Amer. Math. Soc. 330 (1992), 227-256.
7. V.S. Sunder, “On the relation between subfactors and hypergroups,” Applications of hypergroups and related measure algebras, Contemp. Math. 183 (1995), 331-340.
8. N.J. Wildberger, “Duality and entropy for finite abelian hypergroups and fusion rule algebras,” J. London Math. Soc. 56(2) (1997), 275-291.
9. N.J. Wildberger, “Lagrange's theorem and integrality for finite commutative hypergroups with applications to strongly regular graphs,” J. Algebra 182 (1996) 1-37.
2. V.F.R. Jones, “Index for subfactors,” Invent. Math. 72 (1983), 1-15.
3. V.F.R. Jones and V.S. Sunder, Introduction to Subfactors, Cambridge University Press, 1997.
4. Y. Kawada, “ \ddot Uber den Dualitatssatz der Charaktere nichtcommutative Gruppen,” Proc. Phys. Math. Soc. Japan 24(3) (1942), 97-109.
5. N. Obata and N.J. Wildberger, “Generalized hypergroups and orthogonal polynomials,” Nagoya Math. J. 142 (1996), 67-93.
6. V.S. Sunder, “II1 factors, their bimodules and hypergroups,” Trans. Amer. Math. Soc. 330 (1992), 227-256.
7. V.S. Sunder, “On the relation between subfactors and hypergroups,” Applications of hypergroups and related measure algebras, Contemp. Math. 183 (1995), 331-340.
8. N.J. Wildberger, “Duality and entropy for finite abelian hypergroups and fusion rule algebras,” J. London Math. Soc. 56(2) (1997), 275-291.
9. N.J. Wildberger, “Lagrange's theorem and integrality for finite commutative hypergroups with applications to strongly regular graphs,” J. Algebra 182 (1996) 1-37.