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S-Rings, Gelfand Pairs and Association Schemes

  • Kenneth W. Johnson
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2233)

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.

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Authors and Affiliations

  • Kenneth W. Johnson
    • 1
  1. 1.Department of MathematicsPennsylvania State UniversityAbingtonUSA

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