@inbook{8d7693075d9240e0ad7d504e2db356fd,

title = "S-Rings, Gelfand Pairs and Association Schemes",

abstract = "The construction by Frobenius of group characters described in Chap. 1 may be generalized to the case where a permutation group G acting on a finite set has the Gelfand pair property explained below. A general setting which encompasses and extends this is that of an association scheme. For any association scheme there is available a character theory, which in the case where the scheme arises from a group coincides with that of group characters. The development of the theory is described in this chapter. Firstly Schur investigated centralizer rings of permutation groups, then Wielandt defined S-rings over a group. The theory of association schemes provides a character theory even when a group is not present, and this can be applied to obtain a character theory for a loop or quasigroup. In particular a Frobenius reciprocity result was obtained for quasigroup characters, and it was realized that this is available for arbitrary association schemes. A further idea, that of fusion of characters of association schemes, leads to interesting results including a “magic rectangle” condition.",

author = "Johnson, {Kenneth W.}",

year = "2019",

month = jan,

day = "1",

doi = "10.1007/978-3-030-28300-1_4",

language = "English (US)",

series = "Lecture Notes in Mathematics",

publisher = "Springer Verlag",

pages = "137--175",

booktitle = "Lecture Notes in Mathematics",

address = "Germany",

}