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CONTINUOUS ASSOCIATION SCHEMES AND HYPERGROUPS

Part of: Abstract harmonic analysis Algebraic combinatorics Other generalizations of groups

Published online by Cambridge University Press:  27 July 2018

MICHAEL VOIT
MICHAEL VOIT*
Affiliation:
Fakultät Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44221 Dortmund, Germany email michael.voit@math.tu-dortmund.de
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Abstract

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Classical finite association schemes lead to finite-dimensional algebras which are generated by finitely many stochastic matrices. Moreover, there exist associated finite hypergroups. The notion of classical discrete association schemes can be easily extended to the possibly infinite case. Moreover, this notion can be relaxed slightly by using suitably deformed families of stochastic matrices by skipping the integrality conditions. This leads to a larger class of examples which are again associated with discrete hypergroups. In this paper we propose a topological generalization of association schemes by using a locally compact basis space $X$ and a family of Markov-kernels on $X$ indexed by some locally compact space $D$ where the supports of the associated probability measures satisfy some partition property. These objects, called continuous association schemes, will be related to hypergroup structures on $D$. We study some basic results for this notion and present several classes of examples. It turns out that, for a given commutative hypergroup, the existence of a related continuous association scheme implies that the hypergroup has many features of a double coset hypergroup. We, in particular, show that commutative hypergroups, which are associated with commutative continuous association schemes, carry dual positive product formulas for the characters. On the other hand, we prove some rigidity results in particular in the compact case which say that for given spaces $X,D$ there are only a few continuous association schemes.

Keywords

association schemesGelfand pairshypergroupsspherical functionspositive definite functionspositive product formulasrigidity resultsrandom walks on association schemes

MSC classification

Primary: 43A62: Hypergroups
Secondary: 05E30: Association schemes, strongly regular graphs 33C67: Hypergeometric functions associated with root systems 20N20: Hypergroups 43A90: Spherical functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 106 , Issue 3 , June 2019 , pp. 361 - 426
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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